Algebraic K-theory of hyperbolic 3-simplex reflection groups

LO WER ALGEBRAIC K-THEOR Y OF HYP ERBOLIC 3-SIMPLEX REFLECTION GR OUPS. JEAN-FRANC ¸ OIS LAFONT AND IV ONNE J. OR TIZ Abstract. A h yp erbol i c 3-simplex reﬂection group is a Coxete r group arising as a lattice in O + (3 , 1), with fundament al domain a geodesic simplex in H 3 (possibl y with some ideal v ertices). The classiﬁcation of t hese groups is kno wn, and there are exactly 9 co compact examples, and 23 non-cocompact examples. W e prov ide a complete computation of the lo wer algebraic K- theory of the int egral group ri ng of all the h yperb olic 3- simplex reﬂection groups. 1. Intr oduction In this pap er, we pro ceed to g ive a complete computation of the lower a lgebraic K -theory o f the integral gr oup ring o f all the hype r bo lic 3-simplex reﬂec tio n groups. W e now pro ceed to o utline the main steps o f our approa c h. Since the g roups Γ we are considering a re lattices inside O + (3 , 1), fundamental results of F a r rell and Jones [FJ93] imply that the low er alg ebraic K -theory of the integral gr oup ring Z Γ can be computed by calc ula ting H Γ n ( E V C (Γ); K Z −∞ ), a sp eciﬁc generalized equiv ariant homology theory for a mo del for the clas sifying s pace E V C (Γ) of Γ with iso tropy in the fa mily V C of virtually cy c lic subgr oups of Γ. After in tro ducing the gro ups we ar e interested in (see Sectio n 2 ), we then com- bine results from our previo us pap er [LO] with a recent constructio n of L ¨ uck and W eiermann [L W] to obtain the fo llowing e xplicit formula for the homology group ab ov e: K n ( Z Γ) ∼ = H Γ n ( E F I N (Γ); K Z −∞ ) ⊕ k M i =1 H V i n ( E F I N ( V i ) → ∗ ) . In the formula ab ov e, E F I N (Γ) is a model for the classifying s pace for prop er actions, the co llection { V i } k i =1 are a ﬁnite collection o f virtua lly cyclic subgro ups with sp eciﬁc ge ometric prop er ties , and H V i n ( E F I N ( V i ) → ∗ ) are co kernels of certa in relative a ssembly maps. This e x plicit formula is obtained in Section 3. In view of this explicit for mula, our computation reduces to be ing able to: (1) iden tify fo r each o f our groups the corresp onding collectio n { V i } of virtually cyclic subgroups (done in Section 4 ), (2) be able to calcula te the co kernels of the corresp onding relative assembly maps (done in Section 6), and (3) calculate the homology groups H Γ n ( E F I N (Γ); K Z −∞ ). F or the computation o f the homolog y groups , we note that Q uinn [Q u82] ha s de- veloped a sp ectra l sequence for computing the groups H Γ n ( E F I N (Γ); K Z −∞ ). The E 2 -terms in t he sp e c tral sequence can b e c omputed in terms of the low er alge- braic K -theory of the stabilizers o f cells in a C W -mo del fo r the clas sifying space E F I N (Γ). 1 2 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ In Section 5 , we pro ce ed to give, for each o f the ﬁnite subgr oups app earing as a cell stabilizer , a co mputatio n of the lower alg ebraic K -theory . W e return to the sp ectral sequence computation in Section 7, where we ana lyze some of the maps app earing in the computation o f the E 2 -terms for the Quinn spectr a l sequence. In all 32 cases, the sp ectral sequence collaps e s at the E 2 stage, allowing us to co mplete the computations. The reader who is mer e ly in terested in knowing the results o f the computations is invited to consult T able 6 (for the uniform lattices) a nd T a ble 7 (for the non-unifor m lattices). Finally , in the App endix, we provide a “walk through” o f the computations for tw o of the 32 gr oups we consider. Ac kno wledgments The authors would like to thank T om F arr ell, Ian Leary , and Marco V arisco for many helpful co mmen ts on this pro ject. The graphics in this pap er were kindly pro duced by Dennis Burke. The autho rs are par ticularly gra teful to Br uce Magurn for his extensive help with the computations of the alg ebraic K -theory of ﬁnite groups a ppea ring in Section 5 of this pap er . The ﬁrst author’s work on this pro ject was par tially suppo rted by NSF g r ant DMS - 0606 0 02. 2. The Three-dimensional groups A hyperb olic Coxeter n -simplex ∆ n is a n n -dimensio nal geo desic simplex in H n , all of whos e dihedr al ang les ar e subm ultiples of π or zero. W e allow a simplex in H n to b e unbounded with ideal vertices o n the sphere at inﬁnity of H n . It is k nown that such simplices exis t only in dimensions n = 2 , 3 , . . . , 9, a nd that for n ≥ 3 , there are exactly 72 hyperb olic Coxeter simplices up to congr uence, see [JKR T99] and [JK R T02]. A hyperb olic Coxeter n -simplex reﬂection group Γ is the group generated by reﬂections in the sides of a Coxeter n -simplex in hyper b olic n - space H n . W e will call s uch gr oup a hyp erb olic n -simplex gr oup . According to Vin b erg [V67], the asso c ia ted hyperb olic n -simplex gro ups of all but eight of the 72 s implices are a r ithmetic. The nonarithmetic gr oups a re the hy- per bo lic Coxeter tetrahedra g roups [(3 , 4 , 3 , 5 )] [5 , 3 , 6]. [5 , 3 [3] ], [(3 3 , 6)], [(3 , 4 , 3 , 6)], [(3 , 5 , 3 , 6)], and the 5- dimens ional hyperb olic Coxeter group [(3 5 , 4)]. In dimension three, there are 32 h yp erb olic Coxeter tetrahedra groups. Nine of them are c o compact, see Figur e 1, and 23 are nonco compact, s ee Figure 2. L e t us brieﬂy reca ll how the algebra and g eometry of these g roups are enco ded in the Coxeter dia grams. F rom the a lgebraic view p oint, the Coxeter diagram enco des a presentation of the asso ciated gro up Γ as follo ws: asso cia te a genera tor x i to each vertex v i of the Coxeter diagram (hence a ll of our gro ups will come equipp ed with four gener ators, as the Coxeter diagra ms hav e four vertices). F or the relations in Γ, one has: (1) for every vertex v i , one inserts the r elation x 2 i = 1 (2) if tw o vertices v i , v j are not joine d by a n e dg e, one inserts the relatio n ( x i x j ) 2 = 1 (so combined with the prev ious relation, o ne sees that x i and x j commute, generating a Z / 2 × Z / 2), LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 3 (3) if t wo vertices v i , v j are joined by an unlabelle d edge, one inserts the relation ( x i x j ) 3 = 1 (and in particular, the tw o elements x i , x j generate a s ubgroup isomorphic to the dihedral group D 3 ), (4) if tw o vertices v i , v j are joined b y a n edge with label m ij , one inserts the re- lation ( x i x j ) m ij = 1 (and hence, the tw o elements x i , x j generate a subgroup isomorphic to the dihedral group D m ij ). A sp e cial sub gr oup of Γ will be a s ubgroup gener ated by a subset of the genera ting set. Observ e that such a subgr oup will automa tica lly b e a Coxeter group, with a presentation tha t can again b e read oﬀ from the Coxeter dia gram. Spec ia l sub- groups generated by a pair of generato rs will always b e iso morphic to a (ﬁnite) dihedral group. An imp or tant p oint for our purpo ses is that in our Coxeter groups, every ﬁnite subg roup can b e conjugated into a ﬁnite sp ecial subgro up. In par ticular, since ther e are o nly ﬁnitely many sp ecial subgro ups, one can quite easily classify up to iso morphism all the ﬁnite s ubgroups a pp ear ing in any of our 32 Coxeter gro ups . Now let us mov e to the geometric viewp oint. As we mentioned earlier , asso cia ted to any of o ur 32 Coxeter g r oups, o ne ha s a simplex ∆ 3 in hyperb olic 3-space H 3 . Each of the four gener a tors x i of the Coxeter gr oup Γ is bijectively asso ciated with the hyperplane P i extending one of the four fac e s of the simplex ∆ 3 , and the angles betw een the r esp e ctiv e hyper planes ca n again b e read o ﬀ fro m the Coxeter diagra m: (1) if tw o vertices v i , v j are not joined b y an edg e, then ∠ ( P i , P j ) = π / 2 (2) if t wo vertices v i , v j are joined b y an unlab elled edge, then ∠ ( P i , P j ) = π / 3 (3) if tw o vertices v i , v j are joined by an e dge with la b el m ij , then ∠ ( P i , P j ) = π /m ij . The resulting conﬁgur ation of four h yp erplanes exists , and is unique up to iso metries of H 3 . One can now deﬁne the map Γ → O + (3 , 1) = I som ( H 3 ) by se nding each generator x i to the is ometry obtained by r eﬂecting in the corr e spo nding hyper plane P i . The condition on the angles be t ween the hyper pla nes e ns ures that this ma p resp ects the r elations in Γ , and hence is actually a homomor phism. In fa c t this map is an embedding of Γ as a discrete subgr oup of O + (3 , 1), with fundamental domain for the asso ciated actio n on H 3 consisting precisely of the simplex ∆ 3 . Finally , to re la te the geometric with the a lgebraic viewp oint, we re mind the reader o f the following bijective identiﬁcations: (1) giv en an edge in the 3-s implex ∆ 3 , lying on the intersection o f tw o hyper- planes P i , P j , the subgr oup of Γ that ﬁxes the edg e p oint wise is pr ecisely the s pecia l subgroup h x i , x j i (a nd hence will be a dihedral gr oup). (2) giv en a vertex in the 3-s implex ∆ 3 , obtained as the intersection of three hy- per planes P i , P j , P k , the subgro up of Γ that stabilizes the v ertex is precisely the s pecia l subgroup h x i , x j , x k i . W e po int out that the stabilizer of a vertex o f ∆ 3 will either b e a ﬁnite Coxeter group (if the vertex lies inside H 3 ), or will b e a 2-dimensio nal crysta llographic gr oup (if the vertex is an idea l vertex). F urthermor e, one ca n r eadily determine whether a v ertex will b e ideal or not, just by determining whether the asso cia ted sp ecial subgroup is crys tallogra phic or ﬁnite. It is known that for all the groups listed ab ov e the F arr ell and J ones Isomor- phism Conjecture in low er algebra ic K -theory holds , that is H Γ n ( E V C (Γ); K Z −∞ ) ∼ = 4 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ K n ( Z Γ ) for n < 2 (see [Or04, Theo r em 2.1]). Our plan is to use this result to ex- plicitly co mpute the low er a lgebraic K -theor y of the integral group r ing Z Γ, for all of the 32 groups listed ab ov e. • • • • 4 5 [ 4 , 3 , 5 ] • • • • 5 [ 3 , 5 , 3 ] ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ • • • • 5 [ 5 , 3 1 , 1 ] • • • • 4 [ (3 3 , 4) ] • • • • 5 5 [ 5 , 3 , 5 ] • • • • 5 [ (3 3 , 5) ] • • • • 4 4 [ (3 , 4) [2] ] • • • • 5 4 [ (3 , 4 , 3 , 5) ] • • • • 5 5 [ (3 , 5) [2] ] Figure 1. Co compact h yp erb olic Coxeter tetrahedral gr oups 3. A formula f or the algebraic K-theor y. In this se ction, we c ombine so me recent work of L ¨ uc k and W eiermann [L W] with some pr evious work of the a uthors [LO] to establish the following: Prop ositio n 3.1 . L et F ⊂ e F b e a neste d p air of families of su b gr oups of Γ , and assume that the c ol le ction of sub gr oups { H α } α ∈ I is adapte d to the p air ( F , e F ) . L et H b e a c omplete set of r epr esentatives of the c onjugacy classes within { H α } , and c onsider the c el lular Γ -pushouts: ` H ∈H Γ × H E F ( H ) α   β / / E F (Γ)   ` H ∈H Γ × H E e F ( H ) / / X Then X is a mo del for E e F (Γ) . In the ab ove c el lular Γ - pu s hout , we r e quir e either (1) α is the disjoint union of c el lular H -maps ( H ∈ H ), β is an inclusion of Γ - CW-c omplexes, or (2) α is the disjoint union of inclusions of H -CW-c omplexes ( H ∈ H ), β is a c el lular Γ -map. Pr o of. Let us start by recalling tha t a collection { H α } α ∈ I of subg roups of Γ is adapted to the pair ( F , e F ) provided that: (1) F or a ll H 1 , H 2 ∈ { H α } α ∈ I , either H 1 = H 2 , or H 1 ∩ H 2 ∈ F . LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 5 • • • • 6 [ (3 3 , 6) ] • • • • 4 6 [ (3 , 4 , 3 , 6) ] • • • • 5 6 [ (3 , 5 , 3 , 6) ] • • • • 6 6 [ (3 , 6) [2] ] • • • • 5 6 [ 5 , 3 , 6 ] • • • • 6 6 [ 6 , 3 , 6 ] • • • • 6 [ 3 , 3 , 6 ] • • • • 4 6 [ 4 , 3 , 6 ] ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ • • • • [ 3 , 3 [3] ] • • • • 6 [ 3 , 6 , 3 ] ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ • • • • 6 [ 6 , 3 1 , 1 ] ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ • • • • 4 [ 4 , 3 [3] ] ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ • • • • 5 [ 5 , 3 [3] ] ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ • • • • 6 [ 6 , 3 [3] ] • • • • 4 4 [ (3 2 , 4 2 ) ] • • • • 4 4 4 [ (3 , 4 3 ) ] • • • • 4 4 4 4 [ 4 [4] ] ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ • • • • 4 4 [ 3 , 4 1 , 1 ] • • • • 4 4 [ 3 , 4 , 4 ] • • • • 4 4 4 [ 4 , 4 , 4 ] ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ • • • • 4 4 4 [ 4 1 , 1 , 1 ] ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ • • • • [ 3 [3 , 3] ] ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ • • • • [ 3 [ ] × [ ] ] Figure 2. Nonco compact h yp erb olic Co xeter tetra hedral gr o ups (2) The collectio n { H α } α ∈ I is c onjugacy close d i.e. if H ∈ { H α } α ∈ I then g H g − 1 ∈ { H α } α ∈ I for a ll g ∈ Γ. (3) Ev ery H ∈ { H α } α ∈ I is self-normalizing , i.e. N Γ ( H ) = H . (4) F or a ll G ∈ e F \ F , there exists H ∈ { H α } α ∈ I such that G ≤ H . Note that the subgro ups in the collection { H α } α ∈ I are not assumed to lie within the fa mily e F . Using the existence o f the ada pted family { H α } α ∈ I , one can now deﬁne an equiv alence relation on the subgro ups in e F − F a s follows: w e decre e that G 1 ∼ G 2 if there exists an H ∈ { H α } α ∈ I such that G 1 ≤ H and G 2 ≤ H . Note that ∼ is indeed a n equiv a lence r elation: the symmetric prop erty is immediate, while reﬂexivity follows fro m prop erty (4) of a dapted collection, and transitivity c omes from pr op e rty (1) of ada pted collection. F urthermo re this equiv alence relatio n ha s the fo llowing tw o pr o pe r ties: 6 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ • if G 1 , G 2 ∈ e F − F satis ﬁes G 1 ≤ G 2 , then G 1 ∼ G 2 (immediate from the deﬁnition of ∼ ). • if G 1 , G 2 ∈ e F − F a nd g ∈ Γ, then G 1 ∼ G 2 ⇔ g G 1 g − 1 ∼ g G 2 g − 1 (follows from prop erty (2) of a dapted collection). W e deno te by [ e F − F ] the set of equiv a lence classes o f elements in e F − F under the a bove equiv alence r elation, a nd fo r G ∈ e F − F , w e will write [ G ] for the corr e- sp onding equiv a le nce class. Note that by the second prop erty ab ove, the Γ-a ction by conjugation on e F − F preserves equiv alence classes, and hence des cends to a Γ- action on [ e F − F ]. W e let I b e a complete set of r epresentativ es [ G ] of the Γ-or bits in [ e F − F ]. Fina lly , we deﬁne for G ∈ e F − F the subgroup: N Γ [ G ] := { g ∈ Γ | [ g Gg − 1 ] = [ G ] } which is precisely the isotr opy gro up o f [ G ] ∈ [ e F − F ] under the Γ- a ction induced by co njugation. Finally , deﬁne a family of subg roups e F [ G ] of the g roup N Γ [ G ] by: e F [ G ] := { K ⊂ N Γ [ G ] | K ∈ e F − F , [ K ] = [ G ] } ∪ { K ⊂ N Γ [ G ] | K ∈ F } Observe that the no tions deﬁned a bove (intro duced in [L W]) make s ense for any equiv alence relation on e F − F satisfying the tw o pro per ties ab ove. Now [L W, Theorem 2.3 ] states that fo r a ny equiv a lence r elation ∼ on the e le - men ts in e F − F satisfying the tw o pr op erties a bove (and with the nota tion used in the pr evious paragr a ph), the Γ- CW-complex X deﬁned by the cellular Γ-pushout depicted below is a mo del for E e F (Γ). ` [ H ] ∈ I Γ × N Γ [ H ] E F ∩ N Γ [ H ] ( N Γ [ H ]) α   β / / E F (Γ)   ` [ H ] ∈ I Γ × N Γ [ H ] E e F [ H ] ( N Γ [ H ]) / / X In the ab ov e cellular Γ-pushout, L¨ uck-W eiermann requir e either (1) α is the dis- joint union of cellula r N Γ [ H ]-maps ([ H ] ∈ I ), β is an inclusio n of Γ-CW-complexes, or (2) α is the disjo in t union o f inclus io ns of N Γ [ H ]-CW-complexes ([ H ] ∈ I ), β is a cellula r Γ-ma p. W e now pro ceed to verify that, for the equiv a lence re lation we hav e deﬁned using the adapted family { H α } α ∈ I , the left hand ter ms in the cellular Γ-pushout given ab ov e re duce to pr ecisely the le ft hand terms app ear ing in the statement of our prop osition. This boils down to tw o claims: Claim 1: F or any G ∈ e F − F , w e have the equality N Γ [ G ] = H where H is the unique element in { H α } α ∈ I satisfying G ≤ H . T o see this, we ﬁrst note that there indeed is a unique H ∈ { H α } α ∈ I satisfying G ≤ H , for if there were tw o such gr oups H 1 6 = H 2 , then we would immediately see tha t H 1 ∩ H 2 ≥ G ∈ e F − F , contradicting the prop erty (1) of an adapted collection. Next w e observe that if h ∈ H , then h Gh − 1 ≤ hH h − 1 = H , a nd hence that [ hGh − 1 ] = [ G ], which implies the containmen t H ≤ N Γ [ G ]. Conv ersely , if k ∈ N Γ [ G ], then we have that [ G ] = [ k Gk − 1 ], a nd so from the deﬁnition of the eq uiv alence r elation ther e must exist some ¯ H ∈ { H α } α ∈ I with G ≤ ¯ H a nd k Gk − 1 ≤ ¯ H . Since we alr eady k now that G ≤ H , the uniqueness forces ¯ H = H , LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 7 and thus that k Gk − 1 ≤ H . This in turn tells us tha t H ∩ k − 1 H k ≥ G ∈ e F − F , and prop erty (1) of an adapted collection now forces H = k − 1 H k , which implies that k ∈ N Γ ( H ). But pr op erty (3) of a n adapted co llection forces the g roup H to be self-normalizing , giving k ∈ H , and completing the pro of of the reverse inclus io n. Claim 2: F or any G ∈ e F − F , the family e F [ G ] on the gr oup N Γ [ G ] = H (see the pr evious Claim) coincides with the restriction e F ∩ H of the family e F to the subgroup H (i.e. consisting of all elements in e F that lie within H ). Note that the containmen t e F [ G ] ⊂ e F ∩ H is obvious from the deﬁnition of e F [ G ]. F or the oppo site containmen t, let K ∈ e F ∩ H ⊂ e F , and observe that K ≤ H and either K ∈ F , or K ∈ e F − F . In the ﬁrst case , we hav e K ∈ F ∩ H ⊂ e F [ G ], while in the second case, we have that [ K ] = [ G ] by the deﬁnition of the equiv a le nce rela tion, and hence aga in we have K ∈ e F [ G ]. This gives us the c o nt ainment e F ∩ H ⊂ e F [ G ], giving us the Claim. Having establis hed our t wo Cla ims, we can now substitute the expres s ions from the Claims for the c o rresp onding ones in the L ¨ uck-W eiermann diagr am. Finally , we comment on the indices in the disjoint sums app earing in the right ha nd of the diagrams. In the expre ssion of L ¨ uck-W eiermann, the disjoint sum is taken ov er I , a complete sy stem of representatives [ G ] of the Γ-or bits in [ e F − F ]. But observe that from the deﬁnition of the equiv alence re lation we ar e using, cla sses in [ e F − F ] ca n b e bijectiv ely ident iﬁed with gr oups H ∈ { H α } α ∈ I (b y asso ciating each class in [ e F − F ] with the unique element in { H α } α ∈ I containing all the elements in the class ). Since it is clear that the Γ - action on [ e F − F ] coincides (under the bijection above) with the Γ-action on the set { H α } α ∈ I , w e can replace the system of representatives I by the sys tem o f repres en tatives H . This completes the pro of of the prop osition.  W e now sp ecialize to the ca s e wher e F = F I N and e F = V C , and r ecall that Bartels [Ba r03] has established that for any group Γ, the relative a s sembly map: H Γ ∗ ( E F I N (Γ); K Z −∞ ) → H Γ ∗ ( E V C (Γ); K Z −∞ ) is split injective. W e denote by H Γ ∗ ( E F I N (Γ) → E V C (Γ)) the cokernel o f the relative assembly map. Now applying the induction str ucture on this eq uiv ariant generalized homology theory , an immediate co nsequence of the previo us pro po sition is the following: Corollary 3. 2. Given the gr oup Γ , assume that the c ol le ction of sub gr oups { H α } α ∈ I is adapte d t o t he p air ( F I N , V C ) . If H b e a c omplete set of r epr esentatives of t he c onjugacy classes within { H α } , then we have a splitting: H Γ ∗ ( E V C (Γ); K Z −∞ ) ∼ = H Γ ∗ ( E F I N (Γ); K Z −∞ ) ⊕ M H ∈H H H ∗ ( E F I N ( H ) → E V C ( H )) . Next w e r ecall that the authors established in [LO, Theorem 2 .6] that in the ca se where Γ is hyper bo lic relative to a collection of subg roups { H i } k i =1 (assumed to b e pairwise non- conjugate), then the collection o f subgro ups cons isting of: (1) All conjugates of H i (these will b e called p eripher al sub gr oups ). (2) All maximal virtually inﬁnite cyclic subgro ups V such that V * g H i g − 1 , for a ll i = 1 , · · · k , and for all g ∈ Γ. 8 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ is adapted to the pair of families ( F I N , V C ). Applying the previous co r ollary to this sp ecial case, we ge t: Corollary 3.3. Assume that the gr oup Γ is hyp erb olic r elative to the c ol le ction of sub gr oups { H i } k i =1 (assume d to b e p airwise nonc onjugate). L et V b e a c omplete set of r epr esent atives of the c onjugacy classes of maximal virtual ly inﬁnite cyclic sub gr oups inside Γ which c ann ot b e c onjugate d within any of the H i . Then we have a splitting: H Γ ∗ ( E V C (Γ); K Z −∞ ) ∼ = H Γ ∗ ( E F I N (Γ); K Z −∞ ) ⊕ k M i =1 H H i ∗ ( E F I N ( H i ) → E V C ( H i )) ⊕ M V ∈V H V ∗ ( E F I N ( V ) → ∗ ) . The primary e x ample of r elatively hyperb olic gr oups are gro ups Γ a cting with coﬁnite volume (but not c o compactly) on a simply connected Reimannian mani- fold whose sectional curv ature satisﬁes − b 2 ≤ K ≤ − a 2 < 0. Thes e gr o ups a re hyperb olic rela tive to the “cusp gro ups ” , which one can take to b e the subgro ups arising as stabilizers of ideal p oints in the b oundary at inﬁnity o f the Riemannian manifold. W e note that non-uniform lattices in O + ( n, 1) = I som ( H n ) are ex a mples of relatively hyper bo lic groups, and for this class of gro ups, the cusp gr oups a re au- tomatically ( n − 1)-dimensional cr ystallogr aphic gro ups (this is due to the fact that the horospher es have int rinsic g eometry isometric to R n − 1 ). O bserve that 23 of the groups w e are considering (see Figure 2) are non-uniform la ttices in O + (3 , 1), and hence are r elatively hyperb olic groups, relative to a co lle ction of subg r oups, each of which is is o morphic to a 2-dimensional crys tallographic group. In the situation of the groups we are consider ing, the situation is ev en further simpliﬁed by the following obs erv ations : • P ear s on [Pe98] showed that for any 2-dimens ional cry stallogra phic group H , the r elative ass em bly map is an isomo rphism for n ≤ 1, and hence that: H H n ( E F I N ( H ) → E V C ( H )) = 0 for n ≤ 1. • the authors in [LO, Section 3] gav e a general pro cedure for clas sifying the maximal vir tually c y clic subgro ups of Coxeter groups acting o n H 3 . The groups fall into three types, with inﬁnitely many conjugacy classes of type I I and type I I I, and o nly ﬁn itely many co njug a cy cla sses of type I s ubgroups. F urthermor e, the relative assembly ma p is an isomorphism (for n ≤ 1) for all groups of t yp e II and II I. This is discus s ed in more detail in Section 4. • Berko ve, F arrell, J uan-Pineda, and P ears on [BFPP0 0] established that the F arr ell-Jones isomor phism conjecture ho lds for a ll lattices Γ in hyperb olic space (and k ≤ 1), and hence that one has isomorphisms: K n ( Z Γ) ∼ = H Γ n ( E V C (Γ); K Z −∞ ) for a ll n ≤ 1. Combining thes e obser v ations with the previous corollary yields the follo wing: Corollary 3.4. L et Γ ≤ O + (3 , 1) b e any Coxeter gr oup arising as a lattic e (uniform or non-un iform), and let { V i } k i =1 b e a c omplete set of r epr esentatives for c onjugacy LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 9 classes of typ e I maximal virtual ly inﬁnite cyclic sub gr oups of Γ . Then we have, for al l n ≤ 1 , isomorphisms: K n ( Z Γ) ∼ = H Γ n ( E F I N (Γ); K Z −∞ ) ⊕ k M i =1 H V i n ( E F I N ( V i ) → ∗ ) . In the next sections, we will implemen t this cor ollary to compute the low er algebraic K -theory of the integral gr o up ring s of all 32 of the 3-simplex hyper b olic reﬂection groups. 4. Maximal infinite V C ∞ subgr oups. In this section, we pro ceed to classify the maximal inﬁnite virtually cyclic ( V C ∞ ) subgroups arising in our g roups. Let us star t b y brieﬂy recalling s ome of the results from Section 3 of [LO]. Fir st of a ll, for a lattice Γ in O + ( n, 1), inﬁnite V C subg roups are of t wo t yp es: those that ﬁx a s ing le p oint in the bo undary at inﬁnity , and those that ﬁx a pair of points in the b oundary at inﬁnity . W e call subgr oups of the ﬁrst type p ar ab olic , and tho s e of second type hyp erb olic . Note that every parab olic subgro up can b e conjuga ted in to a cusp gr oup; for the purpo se of o ur classiﬁcation, we will ignor e these subgr oups. The subgroups of hyperb olic type automatically stabilize the ge o desic joining the pair of ﬁxed p oints in the b oundary at inﬁnity . F urthermor e the geo des ic they stabilize will pro ject to a p erio dic cur ve in the quotient spa ce H n / Γ. Note that conv ersely , stabilizer s of per io dic geo des ics are inﬁnite V C subgr oups of Γ. This implies that the maximal hyperb olic t yp e inﬁnite V C subg roups of Γ ar e in bijective corresp ondence with stabilizers of p erio dic geo desics. W e now s p ecia lize to the case wher e n = 3 , and Γ is a Coxeter gro up. In this situation, we can subdiv ide the family of per io dic geo desics in to three types: • a geo des ic whose pro jection has no n-trivial intersection with the interior of the p oly hedron H 3 / Γ, which we call t yp e II I. • a g eo desic who se pro jection lies in the b oundary of the polyhedr on H 3 / Γ, but do e s not lie inside the 1-skeleton of H 3 / Γ, which we call t yp e II. • a geo desic whose pro jection lies in the 1 -skeleton of the po lyhedron H 3 / Γ, which we ca ll type I. F or geo desics o f type I I I, it is e a sy to see that the stabilizer of the ge o desic must be isomo rphic to either Z or D ∞ . F or geo desics of type II, the s ta bilizer is always isomorphic to either Z 2 × Z or Z 2 × D ∞ . The main purp os e of this section will be to classify stabilizer s of type I geo desics for all 3 2 groups which oc c ur as hyperb olic 3-simplex re ﬂe c tio n gro ups. W e sta rt by o utlining our a pproach: all the g r oups w e are considering hav e fundamen tal doma in consisting o f a 3-dimensio nal simplex in H 3 (po ssibly with some ideal vertices). So up to conjuga cy , for each of the groups we ar e consider ing , we can have at most six distinct stabilizers of t yp e I geo desic (o ne for e a ch edge in the fundamental domain, fewer in the pre s ence of ideal vertices). B ut this is actually an ov ercount, as one could p otentially hav e a type I geo desic whose pro jection into the fundamen tal domain passes thr ough s everal of the edges . So the ﬁr st step is to understand how many distinct sta bilizers (up to conjugacy) one obtains. Let us ex pla in how one can ﬁnd out the num b er of distinct stabilizer s . Note that, at every (non-ideal) vertex v of our fundamental doma in 3 -simplex in H 3 , we can co nsider a small ǫ - s phere S v centered at v . Now the tessellation of H 3 by 10 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ copies of the fundamen tal doma in induces a tessellatio n of S v by iso metric spherica l triangles. In fact, the tesse lla tion of S v is the one naturally a sso ciated with the sp ecial subg roup of the Coxeter gro up Γ tha t stabilizes the vertex v . Now note that, if w e were to lab el the three edges of the 3- simplex incident to v , we get a corres p onding lab el of the three vertices of a spherical triang le in the tes sellation of S v . One ca n extend this lab eling via reﬂections, b oth for the tesse lla tion of H 3 and the tessellation of S v . Now g iven a p erio dic g eo desic of t yp e I, with a por tio n of the g eo desic pro jecting to the edge e in the 3-simplex, with e adjac en t to the vertex v , one c an easily “read oﬀ ” from the lab eled tessellation of S v which edge extends the ge o de s ic. Indeed, this will b e pick ed up by the la bel of the vertex in the tessella tio n of S v which is antipo dal to the lab eled vertex co rresp onding to e . In this manner, one can e a sily decide the num b er of distinct stabilizers o f type I geo desics that arise for the 32 groups we ar e consider ing. T o recognize the tessellations arising for the v ario us S v , one now notes that the isometry g roup of ea ch of these tes sellations ca n b e obtained by lo oking at the stabilizer Γ v of the vertex v in the gr oup Γ. These stabilizers a re ﬁnite sp ecial sub- groups, of the Coxeter group Γ, g enerated by three o f the four canonical gener ators of Γ. F rom the class iﬁcation of the 32 hyperb olic 3-simplex g roups, it is easy to list out all the ﬁnite parab olic subgro ups we nee d to consider: there are eight of these, namely Z 2 × D 2 , Z 2 × D 3 , Z 2 × D 4 , Z 2 × D 5 , Z 2 × D 6 , [3 , 3] ∼ = S 4 , [3 , 4] ∼ = Z 2 × S 4 , and [3 , 5 ] ∼ = Z 2 × A 5 . The next s tep is to identify the stabilizer s of the corre spo nding g e o desics. In the situation w e ar e cons ide r ing, all the type I geo desics η that app ear have s tabilizer Stab Γ ( η ) acting with fundamental domain an in terv al. In fact, the interv a l can be ident iﬁed with the quotient space η / Stab Γ ( η ) ⊂ H 3 / Γ, which will b e a union of edges in the 1-s keleton of the 3- simplex H 3 / Γ. Hence the group Stab Γ ( η ) c an be ident iﬁed using Bas s -Serre theory: it will b e the fundamental gro up of a graph o f group, where the graph of gr oup consists of a s ingle edg e joining t wo vertices, with edge/vertex gr oups which can b e explicitly found from the tessella tio ns. W e will say that the geo desic (or sometimes the edge in the 1-s keleton) r eﬂe cts at the t wo endpo in t vertices. Indeed, the edge gro up G e will be precisely the stabilizer of one of the edges in η / Stab Γ ( η ) ⊂ H 3 / Γ. On the other hand, the vertex groups G v , G w can b e found by lo oking at each of the tw o endp oint v ertices v , w for η / Stab Γ ( η ), and studying the spherical tessellations of S v , S w . No te that w e are trying to identify e le ments in Γ, which sta bilize the v ertex v (resp ectively w ), and additional ly map the geo desic η through v to itself. In particular , it must map the pair of antipo dal vertices η ± (corresp onding to the incoming/o utgoing η -directions ) in the tessellatio n of S v to themselves. The subgroup G e ⊂ G v can b e iden tiﬁed with the index 2 subgro up consisting o f elements G v which ﬁx b oth of the p o int s η ± . Now ther e is a n obvious map which p ermutes the tw o p oints η + and η − : namely the reﬂection in the equator equidistant fr o m these tw o p oints. But it is not clear that this reﬂection preser ves the tesse lla tion o f S v ; in s ome ca ses, one will need to r eﬂect in the equator , and then rotate by a certain angle alo ng the η ± axis, in order to obtain an e le men t in Γ v . Note that if the r eﬂection in the equa tor pr eserves the tesse lla tion, then w e immediately obtain that G v ∼ = G e × Z 2 . If the reﬂec tio n in the equa tor do es n ot pr eserve the tes s ellation, then we obtain that G v ∼ = G e ⋊ Z 2 . O ne c a n p erform LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 11 the same analy s is at the vertex w , and hence ﬁnd an expressio n for Stab Γ ( η ) as a n amalgama tio n of the groups G v , G w ov er the index 2 subgro ups G e . W e make tw o obse r v ations: ﬁr s t of all, the stabilizer of an e dge will alwa ys be a sp ecial subgroup of Γ, gener ated by a pair o f canonical g enerator s in Γ. In particular, the g roup G e will always be a dihedra l group D k for some k . Now the vertex gro ups ar e o f t wo types: (1) if the reﬂection in the eq ua tor preserves the tessellation, we obtain G v ∼ = Z 2 × D k , or (2) if the reﬂection in the equator do es not preserve the tessella tio n, then one can explicitly read o ﬀ the semi-dir e c t pro duct structure from the tessellation, and in fact it is easy to see that G v ∼ = D 2 k . In the table below, we list out, for ea ch of the ﬁnite para bo lic subgroups we need to consider, the edge s that re ﬂe c t, as well as the cor resp onding G v . Let us expla in the notation us ed in the table: the ﬁrst co lumn gives the v arious ﬁnite parab olic subgroups that o ccur, the s econd column lists the angles that a ppea r in the spherical triangles o f the corr esp onding tes sellation o f S v . The remaining three columns are ordered from smallest angle to larges t, and expresses whether (1) the corr esp onding edge extends (i.e. do es no t reﬂect) a t v , and (2) if it reﬂects , the corr e spo nding subgroup G v . Z 2 × D 2 π / 2 , π / 2 , π / 2 Z 2 × D 2 Z 2 × D 2 Z 2 × D 2 Z 2 × D 3 π / 3 , π / 2 , π / 2 Z 2 × D 3 extends extends Z 2 × D 4 π / 4 , π / 2 , π / 2 Z 2 × D 4 extends extends Z 2 × D 5 π / 5 , π / 2 , π / 2 Z 2 × D 5 extends extends Z 2 × D 6 π / 6 , π / 2 , π / 2 Z 2 × D 6 extends extends S 4 π / 3 , π / 3 , π / 2 extends extends D 4 Z 2 × S 4 π / 4 , π / 3 , π / 2 Z 2 × D 4 D 6 Z 2 × D 2 Z 2 × A 5 π / 5 , π / 3 , π / 2 D 10 D 6 Z 2 × D 2 T able 1: Finite parab ol i c subgroups & lo cal b ehav ior o f edges. F rom the Coxeter diagrams o f the 32 groups we are c o nsidering, we can now read oﬀ quite easily the n umber (up to conjuga cy) of stabilizers o f t yp e I geo desics. W e now pr o ceed to summarize the r esults of this pro cedure, which we list o ut in T ables 2, 3, and 4. W e r emind the rea der tha t, in addition to these subgroups, there will also b e (up to conjugac y ) co un tably inﬁnitely many max imal V C subgroup o f hyperb olic type isomor phic to one of Z , D ∞ , Z 2 × Z , Z 2 × D ∞ (coming fro m stabi- lizers of type II and type I I I geo desics). T he list b elow can be thought of as the “exceptional” maximal V C s ubgroups of hyper bo lic type. Indeed, as we will see in the subsequent sectio ns , these will b e the only maximal V C subgroups of hyperb olic t yp e that will actually co n tribute to the alg ebraic K-theor y of the a m bient gro ups. 4.1. The uniform lattices . There ar e 9 hyperb olic 3-simplex gro ups with funda- men tal domain a compact 3-s implex in H 3 . T he num b er and type of (no n-ﬁnite) stabilizers o f type I g eo desics are listed in the following table: 12 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ Γ Stab Γ [(3 3 , 5)] D 6 ∗ D 3 D 6 , ( D 2 × Z 2 ) ∗ D 2 D 4 (t wice) , D 10 ∗ D 5 D 10 [5 , 3 , 5] D 2 × D ∞ , D 6 ∗ D 3 D 6 , ( D 5 × Z 2 ) ∗ D 5 D 10 (t wice) [(3 3 , 4)] D 4 × D ∞ , D 6 ∗ D 3 D 6 , ( D 2 × Z 2 ) ∗ D 2 D 4 (t wice) [3 , 5 , 3] D 2 × D ∞ , ( D 3 × Z 2 ) ∗ D 3 D 6 (t wice) , D 10 ∗ D 5 D 10 [5 , 3 1 , 1 ] D 2 × D ∞ (t wice) , D 6 ∗ D 3 D 6 , D 10 ∗ D 5 D 10 , ( D 2 × Z 2 ) ∗ D 2 D 4 [4 , 3 , 5] D 2 × D ∞ , (twice) , D 4 × D ∞ , D 6 ∗ D 3 D 6 , ( D 5 × Z 2 ) ∗ D 5 D 10 [(3 , 5) [2] ] D 2 × D ∞ (t wice) , D 6 ∗ D 3 D 6 (t wice) , D 10 ∗ D 5 D 10 (t wice) [(3 , 4 , 3 , 5)] D 2 × D ∞ (t wice) , D 4 × D ∞ , D 6 ∗ D 3 D 6 (t wice) , D 10 ∗ D 5 D 10 [(3 , 4) [2] D 2 × D ∞ (t wice) , D 4 × D ∞ (t wice) , D 6 ∗ D 3 D 6 (t wice) T able 2: Structure of subg roups of co compact groups. 4.2. O ne i deal vertex. W e hav e nine such Coxeter gr oups, namely the groups: [5 , 3 [3] ], [5 , 3 , 6], [3 2 , 4 2 ], [4 , 3 [3] ], [3 , 3 [3] ], [3 , 4 1 , 1 ], [4 , 3 , 6], [3 , 3 , 6], and [3 , 4 , 4]. The nu mber and type of (non-ﬁnite) stabilizers of type I geo des ics, as w ell as the cusp subgroups a re listed in the following table: Γ Stab Γ Cusp [3 , 3 [3] ] 1 edg e D 4 ∗ D 2 D 4 [3 [3] ] [3 , 3 , 6] 1 edg e ( D 2 × Z 2 ) ∗ D 2 D 4 [3 , 6] [5 , 3 [3] ] 2 edg es D 2 × D ∞ , D 10 ∗ D 5 D 10 [3 [3] ] [5 , 3 , 6] 2 edg es D 2 × D ∞ , ( D 5 × Z 2 ) ∗ D 5 D 10 [3 , 6] [(3 2 , 4 2 )] 2 edges D 2 × D ∞ , D 6 ∗ D 3 D 6 [4 , 4] [4 , 3 [3] ] 2 edg es D 2 × D ∞ , D 4 × D ∞ [3 [3] ] [3 , 4 , 4] 2 edg es D 2 × D ∞ , ( D 3 × Z 2 ) ∗ D 3 D 6 [4 , 4] [3 , 4 1 , 1 ] 3 edges D 2 × D ∞ (t wice) , D 6 ∗ D 3 D 6 [4 , 4] [4 , 3 , 6] 3 edg es D 2 × D ∞ (t wice) , D 4 × D ∞ [3 , 6] T able 3: Structure of subg roups of 1-ideal v ertex groups . 4.3. Two i deal vertices. W e hav e nine suc h Coxeter gr oups, namely the groups: [(3 , 5 , 3 , 6)], [(3 , 4 3 )], [(3 , 4 , 3 , 6)], [(3 3 , 6)], [3 [3 , 3] ], [6 , 3 1 , 1 ], [3 , 6 , 3], [6 , 3 , 6], and [4 , 4 , 4]. Note that for these g roups, we have only one edg e segment in the fundament al do- main to consider . LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 13 F or the groups [(3 3 , 6)] and [3 , 6 , 3], the edge extends to one o f the no n-compact edges, a nd hence we again have that there are no p erio dic geo desic s of type I. So in b oth of these cases , we hav e that the maximal virtually inﬁnite V C subgroups of hyperb olic type ar e isomorphic to Z , D ∞ , Z 2 × Z , or Z 2 × D ∞ . In the r emaining cases, the edge r eﬂects at b oth of its endp oints. The stabilizer s we o bta in, as w ell as the t wo cusp subgroups, a re listed out in the following ta ble: Γ Stab Γ Cusp [(3 , 5 , 3 , 6)] D 10 ∗ D 5 D 10 [3 , 6] (t wice) [(3 , 4 3 )] D 6 ∗ D 3 D 6 [4 , 4] (t wice) [(3 , 4 , 3 , 6)] D 4 × D ∞ [3 , 6] (t wice) [3 [3 , 3] ] D 4 ∗ D 2 D 4 [3 [3] ] (t wice) [6 , 3 1 , 1 ] ( D 2 × Z 2 ) ∗ D 2 D 4 [3 , 6] (t wice) [6 , 3 , 6] D 3 × D ∞ [3 , 6] (t wice) [4 , 4 , 4] D 2 × D ∞ [4 , 4] (t wice) T able 4: Structure of subg roups of 2-ideal v ertex groups . 4.4. Three ide al v ertices. W e have t wo such Coxeter gro up: [6 , 3 [3] ] and [4 1 , 1 , 1 ]. Again, there will be no p er io dic geo des ics of type I. So we obtain that the max imal virtually inﬁnite V C s ubgroups of hyperb olic t yp e ar e isomor phic to Z , D ∞ , Z 2 × Z , or Z 2 × D ∞ . 4.5. F our ideal v ertices. There ar e three such Coxeter gro ups, namely the gro ups [3 [ ] × [ ] ], [4 [4] ] and [(3 , 6 ) [2] ]. I t is clear that these gr oups hav e no p erio dic geo desic s of t yp e I, and hence the only max imal virtually inﬁnite V C subgroups of hyperb olic t yp e in b oth of thes e groups are iso mo rphic to Z , D ∞ , Z 2 × Z , or Z 2 × D ∞ . 5. The a lgebraic K-theor y of m aximal finite subgr oups. The only (maximal) ﬁnite subgr oups which o c c ur inside the 32 groups we are int erested in are, up to isomor phis m, one of the following gr oups (see the previous section): Maximal ﬁnite subgroups : 1 , Z / 2, D n for n = 2 , 3 , 4 , 5 , 6 , 10 (here D n denotes the dihedral gr o up of order 2 n ), D 2 × Z / 2, D 6 × Z / 2, S 4 , S 4 × Z / 2, a nd A 5 × Z / 2. Note that these g roups are prec is ely the v arious ﬁnite para bo lic s ubg roups (in the Coxeter s ense) app earing a mongst the 3 2 Coxeter groups we ar e consider ing. In the sp ectra l sequence computing the homology H Γ ∗ ( E F I N (Γ); K Z −∞ ) the E 2 - term is given by the alg ebraic K - groups of the ﬁnite subgroups. The non-tr ivial K -groups ar e listed in T able 5 at the end of this section. F or a ll but four of the ﬁnite groups in o ur list, their low er a lgebraic K -theory is well known. The r elev ant reference a re lis ted b elow for each o f these groups: 14 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ • Z / 2: F or the negative K -g roups, we refer the reader to [C80 a]; the fact that K − 1 ( Z [ Z / 2]) = 0 can also be found in [Bas68, Theorem 10.6, page 695]. The v a nishing of ˜ K 0 can be found in [CuR87 , Corollar y 5.1 7]. F or information abo ut the v anishing of W h we r efer the reader to [O 89]. • D 2 × Z / 2: F o r the nega tive K -g roups, we refer the r eader to [C80a]. The formula in Bas s [Ba s 68, C ha pter 12] shows also that K − 1 ( Z [ D 2 × Z / 2]) = 0. The v anishing o f ˜ K 0 can b e found in [Re76], and for the information concerning W h we refer the r e ader to [Ma78], [Ma 80], [O 89]. • D n , n = 2 , 3 , 4: F or the v anis hing of the negative K -gro ups , we refer the reader to [C80a]. F or the ˜ K 0 we refer the reader to [Re76]. In particular, the v anishing of ˜ K 0 ( Z G ) is prov en for G = D 3 in [Re76, Theorem 8.2 ] and for G = D 4 in [Re76, Theo rem 6.4]. F o r informa tio n ab out W h we r efer the r eader to [Ma78], [Ma80] a nd [O89]. • D 5 : As far as w e know the only K -gro ups found in the literature are ˜ K 0 ( Z D 5 ) ∼ = 0 (see [Re7 6], [EM7 6]) a nd K q ( Z D 5 ) ∼ = 0 for all q ≤ − 2 (see [C80 a]). T o compute K − 1 ( Z D 5 ), we used r esults that ca n b e found in [C80a], a nd [C8 0b], and to compute W h ( D 5 ), we used r e sults that can b e found in [Ma78], [Ma80] and [O89] (see the details in the Section 5.1). • D 10 : As far as we know the o nly K -gro ups found in the literature a re ˜ K 0 ( Z D 10 ) ∼ = 0 (see [Re7 6], [EM7 6]) and K q ( Z D 10 ) ∼ = 0 for all q ≤ − 2 (see [C80 a]). F o r the K − 1 ( Z D 10 ), we used the r esults found in [C8 0a], and [C80b], and for W h ( D 10 ), we used results that ca n b e found in [Ma7 8], [Ma80] and [O89] (see the details in the Section 5.2). • D 6 : See the discussion in Section 5 .1. The whitehead gr oups W h q ( D 6 ) for q ≤ 1 ca n also b e found in [Pe98 , Section 3], and [Or04, Sectio n 5]. • D 4 × Z / 2: Ortiz in [Or04, Section 5] using results fr o m [C80a] [C80b], [CuR87], [O8 9] a nd [Ma06] show e d that K q ( Z [ D 4 × Z / 2]) = 0, q ≤ − 1, ˜ K 0 ( Z [ D 4 × Z / 2]) ∼ = Z / 4, and that W h ( D 4 × Z / 2) is trivial. • S 4 : computed in [BFP P00] • S 4 × Z / 2: computed in [Or 04, Section 5]. F or the remaining gr oups in our list, we deta il the computatio ns in the next few subsections. 5.1. The Lo w er algebraic K -theory of D 6 × Z / 2 . Car ter shows in [C80a] that for q ≤ − 2, K q ( Z F ) = 0 when F is a ﬁnite gr oup. T o calculate K − 1 ( Z [ D 6 × Z / 2]), we use the follo wing form ula due to Carter [C8 0 b, Theor em 3]. Let G be a g r oup of order n , let p denote a prime num b er, let ˆ Z p denote the p -adic integers and let ˆ Q p denote the p - adic num b ers. Then the fo llowing s equence is exa ct: 0 → K 0 ( Z ) → K 0 ( Q G ) ⊕ M p | n K 0 ( ˆ Z p G ) → M p | n K 0 ( ˆ Q p G ) → K − 1 ( Z G ) → 0 . The group alg ebra Q [ D 6 × Z / 2] is isomo rphic to Q 8 × ( M 2 ( Q )) 4 and the same statement is true if Q is replaced by ˆ Q 2 and ˆ Q 3 . Hence K 0 ( Q [ D 6 × Z / 2 ]) ∼ = LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 15 K 0 ( ˆ Q 2 [ D 6 × Z / 2]) ∼ = K 0 ( ˆ Q 3 [ D 6 × Z / 2]) ∼ = Z 12 . The integral p -adic terms are K 0 ( ˆ Z 2 [ D 6 × Z / 2]) ∼ = K 0 ( F 2 [ D 6 × Z / 2] ∼ = K 0 ( F 2 [ D 6 ]) ∼ = Z 2 , (see [O r 04, pag e 350]), and K 0 ( ˆ Z 3 [ D 6 × Z / 2]) ∼ = K 0 ( F 3 [ D 6 × Z / 2]) ∼ = K 0 ( F 3 [( Z / 2) 3 ]) ∼ = Z 8 . Carter a ls o shows in [C8 0a] that K − 1 ( Z [ D 6 × Z / 2 ]) is tor s ion free, so counting ranks in the exact s equence, we hav e that K − 1 ( Z [ D 6 × Z / 2]) ∼ = Z 3 . T o compute ˜ K 0 ( Z [ D 6 × Z / 2]), consider the following Cartesia n square Z [ Z / 2][ D 6 ]   / / Z [ D 6 ]   Z [ D 6 ] / / F 2 [ D 6 ] which yields the May er-Vietories sequence (see [CuR87, Theorem 49.27]) K 1 ( Z [ D 6 × Z / 2]) → K 1 ( Z D 6 ) ⊕ K 1 ( Z D 6 ) ϕ − → K 1 ( F 2 [ D 6 ]) → → ˜ K 0 ( Z [ Z / 2][ D 6 ]) → ˜ K 0 ( Z D 6 ) ⊕ ˜ K 0 ( Z D 6 ) → 0 (1) W e now pro ceed to compute the v arious ter ms app earing in this sequence. W e start by lo oking a t the terms inv olving Z D 6 . In [Re7 6] Reiner shows that ˜ K 0 ( Z D 6 ) is trivial. K 1 ( Z D 6 ) can b e computed as follows: since W h ( G ) equals K 1 ( Z G ) / {± G ab } , the ra nk of K 1 ( Z G ) is equal to the rank o f W h ( G ). But the rank of W h ( G ) is y = r − q , where r denotes the n umber of irreducible r eal r epre- sentations of G , and q denotes the num b er of irreducible r a tional r epresentations of G . In [Bas6 5] Bass shows that r is equal to the num b er of conjugacy clas ses of sets { x, x − 1 } , x ∈ G and q is the num be r of co njugacy classe s of cyclic s ubgroups of G (see a ls o [Mi66]). F or G = D 6 , a direct calculatio ns shows that r = q , a nd hence that W h ( G ) is purely tor sion. Next note that the torsion par t of K 1 ( Z G ) is {± 1 } ⊕ G ab ⊕ S K 1 ( Z G ) (see [W74]), a nd hence we hav e that W h ( D 6 ) = S K 1 ( Z D 6 ). Since Magurn [Ma78] has shown that S K 1 ( Z G ) is trivial, w e see that W h ( D 6 ) is trivial. Since ( D 6 ) ab = ( Z / 2 ) 2 , w e obta in that K 1 ( Z [ D 6 ]) = ( Z / 2 ) 3 . Next we consider the remaining terms in the May er-Vietoris sequence. F or G = D 6 × Z / 2, Ma gurn in [Ma8 0, Co rollary 11] s hows that W h ( D 6 × Z / 2) = 0 (note that the rank of W h ( G × Z / 2) is twice the rank of W h ( G ) since r and q get doubled, see Section 5.2 and 5.3). Since ( D 6 × Z / 2) ab = ( Z / 2) 3 , this yields K 1 ( Z [ D 6 × Z / 2]) = ( Z / 2) 4 . Finally , Ma gurn in [Ma 06, Exa mple 9]) shows that K 1 ( F 2 [ D 6 ]) = ( Z / 2 ) 4 . Substituting all the known terms int o the exact sequence in (1) y ields the following exact s equence: (2) ( Z / 2) 4 σ − → ( Z / 2) 3 ⊕ ( Z / 2) 3 ϕ − → ( Z / 2) 4 → ˜ K 0 ( Z [ Z / 2][ D 6 ]) → 0 . Next, we study the image o f ϕ : K 1 ( Z D 6 ) ⊕ K 1 ( Z D 6 ) → K 1 ( F 2 [ D 6 ]). W e cla im that im( ϕ ) = ( Z / 2) 2 . This can be seen as follows: ﬁrst im( ϕ ) = im( ψ ) where ψ : K 1 ( Z D 6 ) − → K 1 ( F 2 [ D 6 ]) is induced by the ca no nical ring homo morphism Z → F 2 . No te the K 1 ( Z ) is a dir ect summand of K 1 ( Z D 6 ) and iso morphic to Z / 2 ; but this summand g o es to zer o in K 1 ( F 2 [ D 6 ]) since it fa ctors through the fo llowing 16 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ commutativ e square Z / 2 = K 1 ( Z )   / / K 1 ( F 2 ) = 0   K 1 ( Z D 6 ) / / K 1 ( F 2 [ D 6 ]) Since K 1 ( Z D 6 ) = ( Z / 2 ) 3 , this forces dim F 2 (im( ϕ )) ≤ 2 . Now from the exa ct sequences given in (1) and (2), we have tha t: dim F 2 (im( ϕ )) = 2 dim F 2 ( K 1 ( Z [ D 6 ])) − dim F 2 (ker ( ϕ )) = 6 − dim F 2 (im( σ )) . Since dim F 2 (im( σ )) ≤ 4, we see that dim F 2 (im( ϕ )) ≥ 2, which forces im( ϕ ) ∼ = ( Z / 2) 2 . The ex a ct sequence now yields ˜ K 0 ( Z [ Z / 2][ D 6 ]) ∼ = ( Z / 2) 2 . 5.2. The computation of the K -groups K − 1 ( Z D 5 ) , and W h ( D 5 ) . T o co mpute K − 1 ( Z D 5 ), we need Carter’s for m ula for K − 1 , [C80b, Theore m 3 ], the reader is referred to Section 5.1. 0 → K 0 ( Z ) → K 0 ( Q D 5 ) ⊕ M p | n K 0 ( ˆ Z p D 5 ) → M p | n K 0 ( ˆ Q p D 5 ) → K − 1 ( Z D 5 ) → 0 . The group algebr a Q D 5 is isomor phic to Q × Q , and the sa me statement is true if Q is replac ed by ˆ Q 2 (recall that √ 5 / ∈ ˆ Q 2 ). F or p = 5, the gro up a lgebra ˆ Q 5 D 5 ∼ = ( ˆ Q 5 ) 2 × M 2 ( ˆ Q 5 ). Hence K 0 ( ˆ Q 2 [ D 5 ]) ∼ = K 0 ( Q [ D 5 ]) ∼ = Z 2 , and K 0 ( ˆ Q 5 [ D 5 ]) ∼ = Z 3 . Using techniques describ ed in [CuR81, Section 5], we hav e that K 0 ( ˆ Z 5 [ D 5 ]) ∼ = K 0 ( F 5 [ D 5 ]) ∼ = K 0 ( F 5 [ Z / 2]) = K 0 ( F 5 × F 5 ) = Z 2 . Als o K 0 ( ˆ Z 2 [ D 5 ]) ∼ = K 0 ( F 2 [ D 5 ]) ∼ = K 0 ( F 2 × M 2 ( F 2 )) = Z 2 . Carter also shows in [C80a] that K − 1 ( Z [ D 5 ]) is torsion free, so counting ranks a s b efore, we hav e that K − 1 ( Z D 5 ) ∼ = 0. Next, we compute W h ( D 5 ). Rec all that W h ( G ) = Z y ⊕ S K 1 ( Z G ). Magurn in [Ma78] prov es that S K 1 v anishes for all ﬁnite dihedral groups. The rank of the torsion free part is y = r − q (see Sectio n 5.1). Since in D 5 = h r, s | r 5 = s 2 = 1 , srs = r − 1 i , there a r e three cyclic subgr oups mo dulo conjuga cy (the triv ial subg roup { e } , C 2 and C 5 ) and four co njugacy classes of sets { x, x − 1 } (cons isting of { e } , { r, r 4 } , { r 2 , r 3 } and { s , s } ), we see that r = 4 a nd q = 3. This yields W h ( D 5 ) ∼ = Z r − q ∼ = Z . 5.3. The comp utatio n o f the K -groups K − 1 ( Z D 10 ) , and W h ( D 10 ) . Carter shows in [C80a] that for q ≤ − 2, K q ( Z D 10 ) = 0. T o ca lculate K − 1 ( Z D 10 ), ag ain using Car ter’s formula for K − 1 [C80b, Theorem 3], we hav e (see Sectio n 5.1 a nd 5.2): 0 → K 0 ( Z ) → K 0 ( Q D 10 ) ⊕ M p | n K 0 ( ˆ Z p D 10 ) → M p | n K 0 ( ˆ Q p D 10 ) → K − 1 ( Z D 10 ) → 0 The group algebr a Q [ D 5 × Z / 2] is isomo rphic to Q 4 and the same statement is true if Q is repla ced by ˆ Q 2 . F o r p = 5, the g roup a lgebra ˆ Q 5 [ D 5 × Z / 2] ∼ = ( ˆ Q 5 ) 4 × ( M 2 ( ˆ Q 5 )) 2 . Hence K 0 ( Q [ D 5 × Z / 2]) ∼ = K 0 ( ˆ Q 2 [ D 5 × Z / 2]) ∼ = Z 4 , and K 0 ( ˆ Q 5 [ D 5 × Z / 2]) ∼ = Z 6 . The integral p -adic terms are K 0 ( ˆ Z 2 [ D 5 × Z / 2]) ∼ = K 0 ( F 2 [ D 5 × Z / 2]) ∼ = K 0 ( F 2 [ D 5 ]) ∼ = Z 2 (see Section 5 .2), and K 0 ( ˆ Z 5 [ D 5 × Z / 2]) ∼ = K 0 ( F 5 [ D 5 × Z / 2]) ∼ = K 0 ( F 5 [( Z / 2) 2 ]) ∼ = Z 4 . Car ter a lso shows in [C80a] that K − 1 ( Z [ D 10 ]) is to rsion free, so c o un ting ranks in the ex act seq uenc e , we hav e that K − 1 ( Z D 10 ) ∼ = Z . LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 17 Next, we compute W h ( D 10 ). Magurn in [Ma7 8] prov es that S K 1 v anishes for all ﬁnite dihedral g roups. F or D 10 = h r, s | r 10 = s 2 = 1 , sr s = r − 1 i , we hav e four cyclic subgroups (modulo conjuga cy) o f D 10 : the trivial subgroup { e } , h s i , h r 2 i and h r i . On the other hand there ar e six conjugac y class es of sets { x, x − 1 } : { e } , { r, r 9 } , { r 2 , r 8 } , { r 3 , r 7 } , { r 4 , r 6 } , { r 5 , r 5 } and { s, s } . This g ives us r = 6 , q = 4, and hence W h ( D 10 ) ∼ = Z r − q ∼ = Z 2 . 5.4. The Lo w er alge braic K -theory o f A 5 × Z / 2 . Carter shows in [C80a] that for q ≤ − 2, K q ( Z F ) = 0 when F is a ﬁnite gr oup. T o compute W h q ( A 5 × Z / 2) for q ≤ 1 , w e ﬁrs t claim that W h q ( A 5 ) =      Z q = 1 0 q = 0 0 q ≤ − 1 . This can b e s een as follows: by [O 89, Theor e m 1 4.6], we hav e that S K 1 ( Z A 5 ) = 0 . The gr oup A 5 has pr ecisely ﬁve (mutually nonisomor phic) irreducible real repr esen- tation, g iving r = 5 . In A 5 the conjuga cy cla sses of cyclic subg roups ar e repres en ted by the tr ivial s ubgroup { e } , h (12)(34) i , h (123) i , h (1234 5) i giv ing us that q = 4. This forces W h ( A 5 ) ∼ = Z r − q = Z . By [EM76], w e hav e tha t ˜ K 0 ( Z A 5 ) = 0; Dress induc- tion as used in [O89, Theorem 1 1.2] shows that K − 1 ( Z A 5 ) = 0, and by [C80 a] w e hav e that K q ( Z A 5 ) = 0 for q ≤ − 2. Now let us co mpute W h ( A 5 × Z / 2). Magurn in [Ma, Exa mple 5 ] shows that S K 1 ( Z [ A 5 × Z / 2]) = 0. Since r ank( W h ( G × Z / 2)) = 2 rank( W h ( G )) and W h ( A 5 ) ∼ = Z we get W h ( A 5 × Z / 2) ∼ = Z 2 . Next, we compute K − 1 ( Z [ A 5 × Z / 2]). Consider the follo wing Ca r tesian s quare Z [ Z / 2][ A 5 ]   / / Z A 5   Z A 5 / / F 2 [ A 5 ] which yields the May er-Vietories sequence (see [CuR87, Theorem 49.27]) . . . → ˜ K 0 ( Z A 5 ) ⊕ ˜ K 0 ( Z A 5 ) ϕ − → ˜ K 0 ( F 2 [ A 5 ]) → → K − 1 ( Z [ Z / 2][ A 5 ]) → K − 1 ( Z A 5 ) ⊕ K − 1 ( Z A 5 ) → · · · (3) from which we ﬁrst obtain K − 1 ( Z [ A 5 × Z / 2]) ∼ = ˜ K 0 ( F 2 [ A 5 ]). Since F 2 [ A 5 ] ∼ = F 2 × M 4 ( F 2 ), we have that K 0 ( F 2 [ A 5 ]) ∼ = Z 2 , from which it follows that K − 1 ( Z [ A 5 × Z / 2]) ∼ = Z . Next, we claim that ˜ K 0 ( Z [ A 5 × Z / 2]) is trivial. T o see this, let H be a sub- group o f G . F or any lo cally free Z G -mo dule M its re striction to H (denoted b y M H ) is a lo ca lly free Z H -mo dule. The mapping deﬁned b y [ M ] → [ M H ] g ives a homomorphism o f ˜ K 0 ( Z G ) → ˜ K 0 ( Z H ). A group H is hyp er-elementary if H is a s e midirect pro duct N ⋊ P of a cyclic normal subg roup N a nd a subgr oup P of prime or der , wher e ( | N | , | P | ) = 1. Denote H ( G ) the full set of non-conjugate hyper- element ary subgr oups of G . W e shall need 18 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ the fo llowing result due to Swan (see [Sw60]): the map (4) ˜ K 0 ( Z G ) − → Y H ∈H ( G ) ˜ K 0 ( Z H ) is a mo nomorphism for any ﬁnite group G. In par ticular, if ˜ K 0 ( Z H ) = 0 for all H ∈ H , then w e immediately obtain ˜ K 0 ( Z G ) = 0. T o b egin the pro of, we ﬁrst list a full set H ( A 5 ) of non-conjugate hyper-elementary subgroups of A 5 : Z / 2 × Z / 2 , D 3 and D 5 . Note that the hyper- element ary subgroups of G × Z / 2 a re of the for m H or H × Z / 2 for H ∈ H ( G ). In pa rticular, the no n- conjugate hyper -elementary subgroups of A 5 × Z / 2 are: Z / 2 × Z / 2, D 3 and D 5 , ( Z / 2) 3 , D 3 × Z / 2 ∼ = D 6 , and D 5 × Z / 2 ∼ = D 10 . By the results a lready mentioned in Sections 5.1 , 5.2, 5.3 , we hav e ˜ K 0 ( Z H ) = 0 for all H ∈ H ( A 5 × Z / 2). The res ult of Swan on the injectivity of the map in (4 ) immediately implies that ˜ K 0 ( Z [ A 5 × Z / 2]) = 0. Q ∈ V C W h q 6 = 0 , q ≤ − 1 ˜ K 0 6 = 0 W h 6 = 0 D 5 Z D 6 K − 1 ∼ = Z D 4 × Z / 2 Z / 4 D 10 K − 1 ∼ = Z Z 2 D 6 × Z / 2 K − 1 ∼ = Z 3 ( Z / 2) 2 S 4 × Z / 2 K − 1 ∼ = Z Z / 4 A 5 Z A 5 × Z / 2 K − 1 ∼ = Z Z 2 T able 5: Lo wer al gebraic K -theory of subgroups Q ∈ F I N 6. Cokernels o f rela tive assembl y maps f or maximal infinite vir tual l y cycl ic subgroups In view o f Corollar y 3.4, w e will need for our computations the cokernels o f the relative as sembly maps for the v ar io us maximal inﬁnite virtually cyclic s ubg roups of Type I. F r om the tables 2, 3, 4 computed in Section 4, we hav e the following list containing al l the ma x imal inﬁnite virtually cyclic subgro ups that a ppea r in the 32 groups we ar e interested in: Maximal inﬁnite virtually cyclic subgroups: Z , D ∞ , Z × Z / 2, D ∞ × Z / 2 , D n × D ∞ , for n = 2 , 3 , 4 , 5, D 4 ∗ D 2 D 4 , and D 2 × Z / 2 ∗ D 2 D 4 . W e ﬁr st note that, for the g roups in o ur list, the co kernels are known to b e trivial in the following cases: • Z : by work o f Bass [Bas68]. LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 19 • D ∞ : by work of W aldhausen [Wd78]. • Z × Z / 2, and D ∞ × Z / 2: by work of P ears on [Pe98, Section 2]. • D 3 × D ∞ : by work of the authors [LO, Section 4] Finally , the author s hav e also s hown in [LO, Section 4] that for the group D 2 × D ∞ , the cokernels of the relative assembly map for n = 0 , 1 are countably inﬁnite direct sums of Z / 2. The remaining four groups in our list will b e discusse d in the following subse c tio ns. Observe that by a result of F arrell a nd Jones [FJ9 5], the co kernels we are inter- ested in H V n ( E F I N ( V ) → ∗ ) are auto matically trivial for n ≤ − 1. In particula r , we only need to fo cus on the cases n = 0, and n = 1. These cokernels are precisely the elusive Bass, F a rrell, and W aldhausen Nil-g roups. W e are a ble to identify these cokernels exactly , with the exception of the cas e D 4 × D ∞ . F or this group, we conten t ourselves with summarizing what we were a ble to o btain in Subsection 6 .4 . W e summarize the non-trivial cokernels in T a ble 6. V ∈ V C H V 0 ( E F I N ( V ) → ∗ ) 6 = 0 H V 1 ( E F I N ( V ) → ∗ ) 6 = 0 D 2 × D ∞ L ∞ Z / 2 L ∞ Z / 2 D 4 ∗ D 2 D 4 L ∞ Z / 2 L ∞ Z / 2 ( D 2 × Z / 2) ∗ D 2 D 4 L ∞ Z / 2 L ∞ Z / 2 D 4 × D ∞ N il 0 N il 1 T able 6: Cokernels of rel ativ e assembly map for maximal V ∈ V C 6.1. The Low er algebraic K -the o ry of D 5 × D ∞ . First, note that D 5 × D ∞ ∼ = D 10 ∗ D 5 D 10 . As b efore K n ( Z Q ) is zero for n < − 1 (see [FJ95]). Since K − 1 ( Z D 5 ) = 0 (see Sec tion 5.2), and K − 1 ( Z D 10 ) = Z (see Section 5.3), we see tha t for Q = D 10 ∗ D 5 D 10 , w e have K − 1 ( Z Q ) = Z ⊕ Z . F or the remaining K -gr oups, we make us e of [CP 02, Lemma 3.8 ]. F or Q = D 10 ∗ D 5 D 10 , ˜ K 0 ( Z Q ) ∼ = N K 0 ( Z D 5 ; C 1 , C 2 ), wher e C i = Z [ D 10 − D 5 ] is the Z D 5 - bimo dule generated by D 10 − D 5 for i = 1 , 2, (see Sectio n 5.2 and 5.3 for the ˜ K 0 ( Z D n ) for n = 5 , 1 0), and W h ( Q ) ∼ = Z 3 ⊕ N K 1 ( Z D 5 ; C 1 , C 2 ), with C 1 and C 2 as befo r e, (see Section 5.2 and 5 .3 for the W h ( D n ) for n = 5 , 10). The Nil-g roups app earing in these computations ar e the W aldhausen’s Nil-groups. Now by [LO(b)], we know that N K i ( Z D 5 ; C 1 , C 2 ) = 0 for i = 0 , 1, v anishes if and only if the cor resp onding F a rrell Nil-gro up v anishes for the canonica l index t wo subgroup D 5 × Z ⊳ D 5 × D ∞ . Note that in this ca se, the F arrell Nil-group is un twisted, and hence is just the Bass Nil-g roup N K i ( Z D 5 ). But Harmon [Ha87] has shown that for ﬁnite gr oups G of square-fre e order (such as D 5 ), the Bass Nil group N K i ( Z G ) v anishes for i = 0 , 1. W e s ummarize our computations in the 20 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ following: W h q ( D 5 × D ∞ ) =          Z 3 q = 1 0 q = 0 Z 2 q = − 1 0 q ≤ − 2 . 6.2. The Lo w er algebraic K -theory of D 2 × Z / 2 ∗ D 2 D 4 . As b efor e K n ( Z Q ) is zero for n < − 1 (see [FJ95]). Since K − 1 ( Z D 2 ) = 0, K − 1 ( Z [ D 2 × Z / 2]) = 0, and K − 1 ( Z D 4 ) = 0 , we see that for Q = D 2 × Z / 2 ∗ D 2 D 4 , we have that K − 1 ( Z Q ) = 0 . F or the remaining K -gr oups using [CP02, Lemma 3.8], we hav e that fo r Q = D 2 × Z / 2 ∗ D 2 D 4 , ˜ K 0 ( Z Q ) ∼ = N K 0 ( Z D 2 ; A 1 , A 2 ), where A 1 = Z [ D 2 × Z / 2 − D 2 ] is the Z D 2 bi-mo dule ge nerated by ( D 2 × Z / 2) − D 2 , and A 2 = Z [ D 4 − D 2 ] is the Z D 2 bi-mo dule generated by D 4 − D 2 . Similarly , we have that W h ( Q ) ∼ = N K 1 ( Z D 2 ; A 1 , A 2 ), where A 1 , A 2 are the bi-mo dules deﬁned a bove. Now recall that in [LO, Theor em 5.2], the authors es ta blished that (1 ) ˜ K 0 ( Z [ D 2 × D ∞ ]) ∼ = L ∞ Z / 2 a nd (2 ) K 1 ( Z [ D 2 × D ∞ ]) ∼ = L ∞ Z / 2. The computation reduced to showing that the W a ldha usen Nil-groups N K i ( Z D 2 ; A 2 , A 2 ) is isomorphic to a n inﬁnite countable sum of Z / 2 (wher e the bi-mo dule A 2 is deﬁned in the pre v ious paragr aph). This was a c hieved by establis hing (1) the existence of a n injection, a nd (2) the existence of a (diﬀerent) surjection, from the Ba ss Nil-gr oup N K i ( D 4 ) ∼ = L ∞ Z / 2 into the corr esp onding W aldhausen Nil- g roup N K i ( Z D 4 ; A 2 , A 2 ). But the reade r can verify that the arg umen t given in [LO] applies verbatim to the W aldhausen Nil-groups N K i ( Z D 4 ; A 1 , A 2 ) app earing in our pr e sent computation. W e conclude that the lower alg ebraic K -theor y o f D 2 × Z / 2 ∗ D 2 D 4 is given by: W h q ( D 2 × Z / 2 ∗ D 2 D 4 ) =      L ∞ Z / 2 q = 1 L ∞ Z / 2 q = 0 0 q ≤ − 1 . 6.3. The Low er algebraic K -theory of D 4 ∗ D 2 D 4 . As b efore K n ( Z Q ) is zer o for n < − 1 (see [FJ95]). Since K − 1 ( Z D 2 ) = 0 and K − 1 ( Z D 4 ) = 0, we see that for Q = D 4 ∗ D 2 D 4 , w e have that K − 1 ( Z Q ) = 0 . F or the rema ining K -gr o ups, using [C P 02, Lemma 3.8 ], w e have that for Q = D 4 ∗ D 2 D 4 , ˜ K 0 ( Z Q ) ∼ = N K 0 ( Z D 2 ; F 1 , F 2 ), where for i = 1 , 2, F i = Z [ D 4 − D 2 ] is the Z D 2 bi-mo dule generated b y D 4 − D 2 . Similarly , we have that W h ( Q ) ∼ = N K 1 ( Z D 2 ; F 1 , F 2 ), with F 1 and F 2 as b efor e. Now using [LO, Theorem 5.2], we concluded that for Q = D 4 ∗ D 2 D 4 W h q ( Q ) =      L ∞ Z / 2 q = 1 L ∞ Z / 2 q = 0 0 q ≤ − 1 . 6.4. The Lo w er algebraic K -theory of D 4 × D ∞ . The a uthors were unable to obtain a n ex plic it computatio n for this gro up. In this ca se, we hav e that D 4 × D ∞ ∼ = ( D 4 × Z / 2 ) ∗ D 4 ( D 4 × Z / 2), and we are interested in the W aldhause n Nil- groups asso ciated to this splitting. A sp ecial case of recent indep endent work of several authors (including H. Reich, F. Q uinn, J. Davis and A. Ranicki) is that LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 21 this W aldhausen Nil-group is isomorphic to the Ba ss Nil-gro up as so ciated to the canonical index tw o subgroup D 4 × Z ins ide D 4 × D ∞ (a considerable s trengthening of the result of the authors in [LO(b)]). In our ta bles, we denote these g roups by N il 0 = N K 0 ( Z D 4 ) and N il 1 = N K 1 ( Z D 4 ) (the lower Nil g roups v anish by work of F ar rell-Jones [FJ95]). These ab elian gr o ups are known to have the following prop erties: (1) N i l 1 is either trivia l or inﬁnitely gener ated [F77], (2) N i l 0 is inﬁnitely ge ne r ated (see be low), (3) in b oth of these groups , the order of every element divides 8 ([CP 02], [G07]). It is very likely that the gro up N il 1 is also no n-trivial, but we were unable to establish this res ult. In order to see that N il 0 is non- tr ivial, consider the following Cartesian s quare: Z [ Z / 4] ∼ = Z [ a ] /a 4 − 1 = 0   / / Z [ a ] /a 2 − 1 = 0 ∼ = Z [ Z / 2]   Z [ i ] ∼ = Z [ a ] /a 2 + 1 = 0 / / F 2 [ a ] /a 2 − 1 = 0 ∼ = F 2 [ Z / 2] which yields the Cartesian s quare for Z [ D 4 ] = Z [ Z / 4 ⋊ α Z / 2] = Z [ Z / 4] α [ Z / 2]: Z [ Z / 4] α [ Z / 2]   / / Z [ Z / 2][ Z / 2]   Z [ i ] α [ Z / 2] / / F 2 [ Z / 2][ Z / 2] where in Z [ i ] α [ Z / 2], the automorphism α acts v ia α ( i ) = − i . W riting D 2 = Z / 2 × Z / 2 and A = Z [ i ] α [ Z / 2], and applying the N K -functor, this Cartesian square yie lds the May er-Vietoris sequence: N K 2 ( F 2 [ D 2 ]) → N i l 1 → N K 1 ( Z [ D 2 ]) ⊕ N K 1 ( A ) → N K 1 ( F 2 [ D 2 ]) → → N il 0 → N K 0 ( Z [ D 2 ]) ⊕ N K 0 ( A ) → N K 0 ( F 2 [ D 2 ]) Several of the groups a ppea ring in this Mayer-Vietoris se q uence ar e known: the group N K 0 ( F 2 [ D 2 ]) v anishes by [Bas 6 8], while the a uthors hav e previo usly shown [LO] that the g roups N K 1 ( Z [ D 2 ]) a nd N K 0 ( Z [ D 2 ]) a re likewise c ount able inﬁnite sums of Z / 2. F o cusing on the tail end of the May er-Vietoris sequence , and substituting in the expressions we a lready know, we see that: . . . → N il 0 → N K 0 ( A ) ⊕ M ∞ Z / 2 → 0 and non- tr iviality of N il 0 follows from the surjectivity on to the countable inﬁnite sum of Z / 2 . In contrast, fo cus ing on the head of the May er-Vietor is sequence, we see that: N K 2 ( F 2 [ D 2 ]) → N il 1 → N K 1 ( A ) ⊕ M ∞ Z / 2 → N K 1 ( F 2 [ D 2 ]) → . . . Hence to e stablish that N il 1 is non-trivial from this sequence, one would need to either: 22 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ • establish that the ﬁrst map is non-zero, i.e . understand the map N K 2 ( F 2 [ D 2 ]) → N K 1 ( Z D 4 ) • establish that the seco nd map is non-zero by showing that the thir d map ha s a no n-trivial k ernel, for ins tance b y understanding the map N K 1 ( Z [ D 2 ]) → N K 1 ( F 2 [ D 2 ]) The authors have some pa rtial results co ncerning some of the terms showing up in the head of the Mayer-Vietoris seq uence, but so far hav e b een unsuccessful in establishing non-triviality of N il 1 . 7. The s pectral sequences and final comput a tion s W e now pro ceed to apply Co r ollary 3.4 to compute the low er alge br aic K-theory of Z Γ, for Γ o ne of the 32 p oss ible 3 -simplex hyperb olic r e ﬂection gro ups . Let us recall that Corolla ry 3.4 tells us that for such groups Γ, we have for n ≤ 1 an isomorphism: K n ( Z Γ) ∼ = H Γ n ( E V C (Γ); K Z −∞ ) ⊕ k M i =1 H V i n ( E F I N ( V i ) → ∗ ) where { V i } k i =1 are a co mplete set of represe n tatives for the c o njugacy classes of maximal inﬁnite virtually cyclic s ubgroups of Type I. W e ﬁrst note that for a ll 32 of our groups , we have: • obtained in Sectio n 4 a complete list of the Typ e I ma ximal inﬁnite virtually cyclic subgroups (listed out in T a bles 2, 3, and 4). • computed in Section 6 the gro ups H V n ( E F I N ( V ) → ∗ ) for all the Type I maxima l inﬁnite virtua lly cyclic subgr oups that o ccur, with the exception of the case V = D 4 × D ∞ . In particula r, this allows us to determine the expression k M i =1 H V i n ( E F I N ( V i ) → ∗ ) o ccurring in the formula ab ove for all 3 2 o f our g roups. Hence we are left with computing H Γ n ( E V C (Γ); K Z −∞ ) for each of our 32 gro ups. In order to do this, we reca ll that Quinn [Q u82] established the existence of a sp ectral s equence which co n verges to this homo logy gro up, with E 2 -terms g iven by: E 2 p,q = H p ( E F I N (Γ) / Γ ; { W h q (Γ σ ) } ) = ⇒ W h p + q (Γ) . The complex that gives the homology o f E V C (Γ) / Γ with lo ca l co eﬃcients { W h q (Γ σ ) } has the form · · · → M σ p +1 W h q (Γ σ p +1 ) → M σ p W h q (Γ σ p ) → M σ p − 1 W h q (Γ σ p − 1 ) · · · → M σ 0 W h q (Γ σ 0 ) , where σ p denotes the cells in dimension p , and the sum is ov er all p -dimensional cells in E V C (Γ) / Γ. T he p th homology gr oup of this complex will give us the e n tries LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 23 for the E 2 p,q -term o f the spec tr al seq uence. Let us reca ll that W h q ( F ) =      W h ( F ) , q = 1 ˜ K 0 ( Z F ) , q = 0 K q ( Z F ) , q ≤ − 1 . Observe that for the groups we are interested it is particularly easy to obtain a mo del for E F I N (Γ) with the additional nice prop erty that the Γ-ac tion is co compact. Indeed, in the case of the 9 co compact lattices, one can just ta ke the Γ- space to b e H 3 , with fundamen tal domain a 3-dimensional simplex. In the case of the 23 no n- uniform lattices, one can Γ-equiv aria nt ly remov e a disjoint collectio n o f horoballs from H 3 to form a Γ-spa ce X Γ on whic h Γ acts co compactly . A fundament al domain for the Γ-a c tion o n the space X Γ can b e o btained by (1) taking the 3- simplex fundamental domain ∆ 3 Γ for the Γ- action on H 3 , and (2) r emoving a small neighborho o d of each idea l v ertex in ∆ 3 Γ . F or the resulting fundamental domain X Γ / Γ, it is particularly easy to iden tify the stabilizers o f ea ch cell. Indeed, there will alwa ys be a single 3-dimensio nal ce ll, with tr ivial iso tropy . The 2-dimensional cells will consist o f (1) precisely four cells corresp onding to the or iginal faces of ∆ 3 Γ , each of which will ha ve stabilizer Z / 2, (2) one additional cell fo r each idea l vertex (obtaine d from “trunca ting” the vertex), with tr ivial sta bilizer. Note that since W h q (1) and W h q ( Z / 2) v anish for all q ≤ 1 , this in particular implies tha t ther e will never b e any c ontribution to the E 2 -terms fr om the 3 - dimensional and 2 -dimensional c el ls. In o ther words, E 2 p,q = 0 except p ossibly for p = 0 , 1. Now let us fo cus on the 1- dimensional and 0 -dimensional ce lls in the fundamen tal domain X Γ . The 1 -dimensional cells will consist of (1) precisely six edges, corres po nding to the original edg es of ∆ 3 Γ , each of which will ha ve stabilizer a dihedra l group D n ( n = 2 , 3 , 4 , 5 , o r 6). (2) three new edges for ea ch ideal vertex (obtained from “ tr uncating the ver- tex), with stabilizer Z / 2. Note that a mongst these g roups, the only o ne s that have some non-trivial W h q are the groups D 5 (for q = 1) a nd D 6 (for q = − 1 ). Now the 0-dimensio nal cells that o ccur c o nsist of (1) one v ertex for each of the non-ideal vertices in ∆ 3 Γ . Each of these vertices will have stabilize r a spherical Coxeter gro up, isomorphic to the sp ecial subgroup of the Coxeter g roup Γ which corres po nds to the vertex (up to isomorphism, these ar e the groups o cc ur ing in T able 1). (2) one new vertex for each edge leading int o an ideal vertex (obtained fro m “truncating the cusp” ). The s tabilizer of the vertex will coincide with the stabilizer o f the corres po nding edge (and hence b e a dihedral gro up D n where n = 2 , 3 , 4 , 5 , o r 6) F or all these gro ups, the non-v anishing W h q can be found in T a ble 5 . Finally , we observe that s ince the o nly 1 -cells with non-tr ivial W h q are the g roups D 5 and D 6 , most of the morphisms in the chain complex for the E 2 -terms will either be zero (or in a few cases, will clear ly b e isomor phis ms). The three morphisms one needs to take c are w ith are: 24 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ • K − 1 ( Z D 6 ) → K − 1 ( Z [ D 6 × Z / 2]), • W h ( D 5 ) → W h ( D 10 ), • W h ( D 5 ) → W h ( A 5 × Z / 2). W e pro ceed to analyze each of these three morphisms in the next three sections. 7.1. The m ap K − 1 ( Z D 6 ) → K − 1 ( Z [ D 6 × Z / 2]) . W e start by obser v ing that K − 1 ( Z D 6 ) ∼ = Z and K − 1 ( Z [ D 6 × Z / 2 ]) ∼ = Z 3 (see T able 5). W e claim that the ma p induced by the natural inclusion D 6 ֒ → D 6 × Z / 2 is injective, and the quotient group is isomorphic to Z 2 . In or der to see this, we merely note that there is a re traction from D 6 × Z / 2 to the subgroup D 6 , and hence we must have that K − 1 ( Z D 6 ) ∼ = Z is a summand inside K − 1 ( Z [ D 6 × Z / 2]) ∼ = Z 3 , whic h immediately gives our claim. 7.2. The m ap W h ( D 5 ) → W h ( D 10 ) . W e start b y observing that W h ( D 5 ) ∼ = Z and W h ( D 10 ) ∼ = Z 2 (see T able 5 ). W e claim that the map induced by the natural inclusion D 5 ֒ → D 10 ∼ = D 5 × Z / 2 is injectiv e, and the q uotient group is isomo rphic to Z . But again, we see that there is a retraction from D 5 × Z / 2 to the subgroup D 5 , and he nc e W h ( D 5 ) ∼ = Z is a summand inside W h ( D 10 ) ∼ = Z 2 , whic h gives us our claim. Note that this map was us e d implicitly in Section 6 .1 (in the argument men tioned in the second paragra ph). 7.3. The map W h ( D 5 ) → W h ( A 5 × Z / 2) . W e start b y observing that W h ( D 5 ) ∼ = Z and W h ( A 5 × Z / 2) ∼ = Z 2 (see T able 5). W e cla im that the map induced by the natural inclusion D 5 ֒ → A 5 × Z / 2 is injective, and the quotient g roup is isomo rphic to Z . Note that in this ca se we do not hav e a retraction from the gro up A 5 × Z / 2 to the s ubgroup D 5 (since A 5 is simple, the only pos s ible non-trivial quotients would be isomo rphic Z / 2, A 5 , or A 5 × Z / 2). Let us start by obser ving that, fr om the inclusio n D 5 ֒ → A 5 , we obtain that the inclusion D 5 ֒ → A 5 × Z / 2 facto r s thro ugh: D 5 ֒ → D 5 × Z / 2 ∼ = D 10 ֒ → A 5 × Z / 2 which implies the map on Whitehead groups likewise facto rs through: W h ( D 5 ) → W h ( D 10 ) → W h ( A 5 × Z / 2) . Observe that the ﬁrst map in the ab ove sequence was analyz ed in the previous Section 7.2. F ur thermore the last tw o gr oups in this sequence are abstrac tly iso - morphic to Z 2 . So in o rder to o btain our claim, all we need to do is establis h that the inclus ion D 10 ֒ → A 5 × Z / 2 induces an isomorphism on Whitehead groups. In o rder to do this, w e recall that Dress induction provides us with an is o mor- phism (see [O89, Chapter 11]): W h ( A 5 × Z / 2) ∼ = lim − → H ∈H ( A 5 × Z / 2) W h ( H ) . Here H ( A 5 × Z / 2 ) co nsists of all h yp erelementary subg r oups of A 5 × Z / 2 , the limit is over all maps induced b y inclusion and co njugation, and the iso morphism is nat- urally induced by the inclusions. Now re c a ll (Section 5.4) that the hyperelementary subgroups o f A 5 × Z / 2 are, up to isomorphism: ( Z / 2 ) 2 , ( Z / 2) 3 , D 3 , D 5 , D 6 , and D 10 . Amongst these groups (se e T able 5), the o nly gr oups with non-trivial W h are the groups D 5 and D 10 , with W h ( D 5 ) ∼ = Z and W h ( D 10 ) ∼ = Z 2 . F ur ther more, inside the group A 5 × Z / 2, it is e a sy to see that: (1) ev ery subg roup isomo rphic to D 5 lies inside a subgr o up isomorphic to D 10 , (2) all the subgro ups isomorphic to D 10 are pairwise co njugate. LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 25 This immediately implies that the direct limit to the rig ht is canonically isomorphic to W h ( D 10 ), which giv es us our desired cla im. Γ K − 1 6 = 0 ˜ K 0 6 = 0 W h 6 = 0 [3 , 5 , 3] Z 4 L ∞ Z / 2 Z 3 ⊕ L ∞ Z / 2 [5 , 3 , 5] Z 4 L ∞ Z / 2 Z 6 ⊕ L ∞ Z / 2 [(3 3 , 4)] Z 2 ( Z / 4) 2 ⊕ L ∞ Z / 2 ⊕ N il 0 L ∞ Z / 2 ⊕ N il 1 [5 , 3 1 , 1 ] Z 2 L ∞ Z / 2 Z 3 ⊕ L ∞ Z / 2 [4 , 3 , 5] Z 3 ( Z / 4) 2 ⊕ L ∞ Z / 2 ⊕ N il 0 Z 3 ⊕ L ∞ Z / 2 ⊕ N il 1 [(3 3 , 5)] Z 2 L ∞ Z / 2 Z 3 ⊕ L ∞ Z / 2 [(3 , 5) [2] ] Z 4 L ∞ Z / 2 Z 6 ⊕ L ∞ Z / 2 [(3 , 4) [2] ] Z 4 ( Z / 4) 4 ⊕ L ∞ Z / 2 ⊕ N il 0 L ∞ Z / 2 ⊕ N il 1 [(3 , 4 , 3 , 5)] Z 4 ( Z / 4) 2 ⊕ L ∞ Z / 2 ⊕ N il 0 Z 3 ⊕ L ∞ Z / 2 ⊕ N il 1 T able 6: The low er algebraic K -theory of the co c ompact hy p erb o lic 3-simpl ex groups 7.4. The sp ectral sequences. By this point of the pap er we have: • describ ed a simple mo del for E F I N (Γ) for our g roups, and identiﬁed the stabilizers of cells (in this se c tio n) • computed (in Section 5 ) the low er algebra ic K -groups of the stabilizers o f the c e lls, and • iden tiﬁed (in this section) the non-trivial morphisms app earing in the co m- putation of the E 2 -terms of the Quinn sp ectr a l sequence . F urthermor e, as explained earlier, the only p ossible non-zero terms in the sp ectral sequence are the E p,q with p = 0 , 1. This bo ils down to understa nding the ho mology of the complex: 0 → M σ 1 W h q (Γ σ 1 ) → M σ 0 W h q (Γ σ 0 ) → 0 , But we’ve seen in this s ection that the middle map is alwa ys injective, hence the E 1 ,q terms will also v anish. This gives us that in al l 32 c ases the sp e ctra l se quenc e c ol la pses at the E 2 -term . In fact, in all 32 cas es, the o nly p ossible non-zer o E 2 -terms are E 2 0 , − 1 , E 2 0 , 0 , E 2 0 , 1 . In par ticular the K i ( Z Γ) v anis h for i ≤ − 2. The results o btained for K − 1 , ˜ K 0 , and W h for all 32 hyper bo lic 3-simplex gr oups are listed o ut in T able 6 a nd T able 7 . F o r ea se of notatio n, we have only entered the non-zer o ter ms in the T ables; a ll the blank squar es r epresent entries where the corres p onding gr oup v anishes. 26 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ Γ K − 1 6 = 0 ˜ K 0 6 = 0 W h 6 = 0 [3 [ ] × [ ] ] [4 [4] ] [(3 , 6) [2] ] Z 2 [6 , 3 [3] ] Z 2 [4 1 , 1 , 1 ] [(3 , 5 , 3 , 6)] Z 3 Z 3 [(3 , 4 3 )] Z 2 ( Z / 4) 2 [(3 , 4 , 3 , 6)] Z 3 ( Z / 4) 2 ⊕ N il 0 N il 1 [(3 3 , 6)] Z [3 [3 , 3] ] L ∞ Z / 2 L ∞ Z / 2 [6 , 3 1 , 1 ] Z L ∞ Z / 2 L ∞ Z / 2 [3 , 6 , 3] Z 3 [6 , 3 , 6] Z 6 ( Z / 2) 4 [4 , 4 , 4] ( Z / 4) 2 ⊕ L ∞ Z / 2 L ∞ Z / 2 [5 , 3 [3] ] Z 2 L ∞ Z / 2 Z 3 ⊕ L ∞ Z / 2 [5 , 3 , 6] Z 5 L ∞ Z / 2 Z 3 ⊕ L ∞ Z / 2 [(3 2 , 4 2 )] Z 2 ( Z / 4) 2 ⊕ L ∞ Z / 2 L ∞ Z / 2 [4 , 3 [3] ] Z 3 ( Z / 4) 2 ⊕ L ∞ Z / 2 ⊕ N il 0 L ∞ Z / 2 ⊕ N il 1 [3 , 3 [3] ] L ∞ Z / 2 L ∞ Z / 2 [3 , 4 1 , 1 ] Z 2 ( Z / 4) 2 ⊕ L ∞ Z / 2 L ∞ Z / 2 [4 , 3 , 6] Z 4 ( Z / 4) 2 ⊕ L ∞ Z / 2 ⊕ N il 0 L ∞ Z / 2 ⊕ N il 1 [3 , 3 , 6] Z 4 L ∞ Z / 2 L ∞ Z / 2 [3 , 4 , 4] Z 2 ( Z / 4) 2 ⊕ L ∞ Z / 2 L ∞ Z / 2 T able 7: The low er algebraic K -theory of the no n- co compact h yp erb oli c 3- s implex groups LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 27 Note that several of the g roup app ear ing in T ables 6 and 7 inv olve copies o f the Bass Nil-groups N K 0 ( Z D 4 ) and N K 1 ( Z D 4 ) (see Section 5.5). In or der to simplify the notation in the ta bles, we will use N il 0 and N il 1 to denote these tw o Nil-gro ups. Recall that we k now that the gr oups N il 0 , N il 1 are torsio n gro ups, where the or de r of every element divides 8, a nd furthermore the gr oup N il 0 is inﬁnitely gene r ated (see Section 5.5). 8. Appendix: tw o specific examples. In this App endix w e work thr ough the entire pro cedur e for t wo sp eciﬁc exam- ples (one co compac t, and one no n-co compact), with a view o f helping the r e ader understand the lay out of the paper . 8.1. The group [(3 , 5) [2] ] . The Coxeter diag r am for this group Γ can be found in Figure 1, from whic h the following presentation can be rea d oﬀ (see Sectio n 2): h w, x, y , z | w 2 = x 2 = y 2 = z 2 = 1 , ( wx ) 3 = ( xy ) 5 = ( y z ) 3 = ( z w ) 5 = ( wy ) 2 = ( xz ) 2 = 1 i . This g roup acts on H 3 co compactly , with fundamental domain a 3-simplex ∆ 3 . After lab eling the hyperplanes extending the four faces by the four gener ators of Γ, the angles b etw ee n these hyperpla nes sa tisfy the following rela tio nships (see Section 2): • ∠ ( P w , P y ) = ∠ ( P x , P z ) = π / 2 , • ∠ ( P w , P x ) = ∠ ( P y , P z ) = π / 3 , • ∠ ( P w , P z ) = ∠ ( P x , P y ) = π / 5 . In particular, the a ction o f Γ on H 3 gives a co compact mo del for E F I N (Γ), and the splitting for m ula (see Co rollary 3 .4) tells us that we hav e, for all n ≤ 1 , isomorphisms: K n ( Z Γ) ∼ = H Γ n ( E F I N (Γ); K Z −∞ ) ⊕ k M i =1 H V i n ( E F I N ( V i ) → ∗ ) . Let us now ident ify the (ﬁnitely many) g roups { V i } that a pp ear in the ab ov e formula. As explained in Section 4, these gro ups will ar ise as stabilizers of T yp e I geo desics, which are precisely (up to the Γ-a ction) one of the six geo desics P w ∩ P x , P w ∩ P y , P w ∩ P z , P x ∩ P y , P x ∩ P z , and P y ∩ P z . T o identif y the stabilizers of these geo desics, we ﬁrs t need to identify the vertex stabilizers for the simplex ∆ 3 . Re c all that these will b e the s pecia l subgroups g enerated by triples of genera tors. But from the Coxeter diagram for Γ, one immediately sees that any triple of vertices spans out a subdia gram co rresp onding to the C oxeter g roup [3 , 5]. This implies that every vertex ha s stabilizer isomorphic to the (ﬁnite) Coxeter group [3 , 5], which is well known to b e isomorphic to the group A 5 × Z / 2. Now for each of the six type I geo desics we hav e, one can co nsider the pro jection to the fundamen tal domain ∆ 3 . F rom T able 1, lo oking up the vertex stabilizer s A 5 × Z / 2, we see that every one of the six geo desics pro jects to precisely the asso ciated edge in ∆ 3 . Now to ﬁnd the stabilizers of the geo de s ics, one applies Ba ss-Serre theo ry . The stabilizer acts on each of the geo desics with quotien t a segment, so one can write ea ch of the stabilizers as a gener a lized free pro duct. F urthermor e, T able 1 allows us to ident ify the vertex gr oups in the Ba ss-Serre gr aph of groups. 28 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ Let us see ho w this works, for instance in the ca se of the geo desic P x ∩ P y . T he t wo a sso ciated hyperplanes P x and P y int ersect a t an angle of π / 5 , hence the edge group in the Bass-Serr e graph of groups will b e D 5 . F o r the vertex groups, we see that the cor resp onding segment in ∆ 3 joins a pa ir o f vertices w ith stabilizer A 5 × Z / 2, and cor resp ond to the a ngle of π / 5 at bo th the vertices. The last r ow in T able 1 tells us that both the vertex groups in the Bass - Serre gr aph of groups will b e D 10 . T his tells us that the stabilizer of the geo desic P x ∩ P y is precisely the group D 10 ∗ D 5 D 10 ∼ = D 5 × D ∞ . Car rying this pro cedure o ut for each o f the six geo desics, o ne ﬁnds that the stabilizers one o btains ar e: • t wo copies of D 10 ∗ D 5 D 10 , cor resp onding to the tw o geo desics P x ∩ P y and P w ∩ P z , • t wo co pies of D 6 ∗ D 3 D 6 , cor resp onding to the tw o geo desics P w ∩ P x and P y ∩ P z , • t wo co pies of D 2 × D ∞ , corres po nding to the t wo geo desics P w ∩ P y and P x ∩ P z . Note that these are precisely the groups that are listed out in T a ble 4. Finally , amongst these six subgroups, one needs to know which o nes have a non-trivia l cokernel for the relative ass em bly map. But from the work in Section 6, all the non-trivial cokernels are listed out in T able 6. Lo o king up T a ble 6, one sees that out of these s ix gr oups, the o nly ones with no n-trivial cokernels ar e the t wo copies of D 2 × D ∞ , each of whom co n tributes L ∞ Z / 2 to the K 0 ( Z Γ) and W h (Γ). So w e are ﬁnally left with computing the homology coming from the ﬁnite sub- groups, i.e. the term H Γ n ( E F I N (Γ); K Z −∞ ). As we men tioned ea r lier, a co compact fundamen tal domain for H 3 / Γ is given by ∆ 3 . The s tabilizers of cells in the funda- men tal doma in can b e read oﬀ from the Coxeter dia g ram, as they will precisely be the s pecia l subgr oups (see the discussion in Section 7). W e see that: • there is one 3-dimensional cell (the interior o f ∆ 3 ), with trivial stabilizer, • there a re four 2-dimensio na l cells (the faces of ∆ 3 ), with s ta bilizer Z / 2 , • there are six 1-dimensio nal cells (the edges of ∆ 3 ), tw o of which hav e stabi- lizer D 2 , t wo o f which hav e s tabilizer D 3 , and tw o of which have stabilizer D 5 , • there ar e four 0- dimensional cells (the vertices of ∆ 3 ), each of which has stabilizer A 5 × Z / 2. Now to obtain the E 2 -terms in the Q uinn sp ectral seq ue nc e , we need the homology of the complex: · · · → M σ p +1 W h q (Γ σ p +1 ) → M σ p W h q (Γ σ p ) → M σ p − 1 W h q (Γ σ p − 1 ) · · · → M σ 0 W h q (Γ σ 0 ) , where σ p are the p - dimensional cells (which we identiﬁed ab ov e). But fro m the work in Section 5, we know explicitly all the groups app ear ing in the a bove complex. Indeed, lo oking up the non-zero K -groups in T a ble 5, we s ee that for q < − 1, the ent ire complex is ident ically zero. F or the remaining v alues o f q , we hav e : q = − 1 : The co mplex degenerates to 0 → 4 K − 1 ( Z [ A 5 × Z / 2]) → 0 , LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 29 where the four copies of K − 1 ( Z [ A 5 × Z / 2]) come from the four vertices of ∆ 3 . Since we know (see T able 5) that K − 1 ( Z [ A 5 × Z / 2]) ∼ = Z , we immediately get that E 2 p, − 1 all v anish, with the exception of E 2 0 , − 1 ∼ = Z 4 . q = 0 : The complex is identically zero, a nd hence we see that E 2 p, 0 all v anish. q = 1 : The complex degener ates to: 0 → 2 W h ( D 5 ) → 4 W h ( A 5 × Z / 2) → 0 . Note that the ﬁrst co py of W h ( D 5 ) comes fro m the edge P x ∩ P y ∩ ∆ 3 , while the second co p y of W h ( D 5 ) comes from the P w ∩ P z ∩ ∆ 3 . The four copies of W h ( A 5 × Z / 2) come from the four v ertices of ∆ 3 . Since the t wo edges P x ∩ P y ∩ ∆ 3 and P w ∩ P z ∩ ∆ 3 are disjoint , the co mplex splits as a sum of t wo sub complexes , one for each o f the tw o edg es. F ocus ing on the ﬁr st edge, we se e that w e hav e: 0 → W h ( D 5 ) → 2 W h ( A 5 × Z / 2) → 0 W e know that W h ( D 5 ) ∼ = Z and W h ( A 5 × Z / 2) ∼ = Z 2 (see T able 5), and that the map W h ( D 5 ) ֒ → W h ( A 5 × Z / 2) induced by inclusion is split injective (see Sectio n 7.3). This immediately tells us that in the chain complex ab ov e, we have that 2 W h ( A 5 × Z / 2 ) /W h ( D 5 ) ∼ = Z 3 . An identical analysis for the o ther edge g ives us that the homology of the origina l complex y ields E 2 1 , 1 ∼ = 0 a nd E 2 0 , 1 ∼ = Z 6 . Combining everything we’v e s a id s o far, we s ee that for the Quinn spectr al sequence, the only non-zer o E 2 -terms are E 2 0 , − 1 ∼ = Z 4 and E 2 0 , 1 ∼ = Z 6 . This implies that the sp ectral sequence immediately collapse s , giving us that H Γ n ( E F I N (Γ); K Z −∞ ) ∼ = 0 for n < − 1 , n = 0, and H Γ − 1 ( E F I N (Γ); K Z −∞ ) ∼ = Z 4 , H Γ 1 ( E F I N (Γ); K Z −∞ ) ∼ = Z 6 . W e now have b oth the terms app ear ing in the splitting formula, and we conclude that the low er algebr a ic K -theory o f the gro up Γ is given by: W h n (Γ) =          W h (Γ) ∼ = Z 6 ⊕ L ∞ Z / 2 , n = 1 ˜ K 0 ( Z Γ) ∼ = L ∞ Z / 2 , n = 0 K − 1 ( Z Γ) ∼ = Z 4 , n = − 1 K n ( Z Γ) ∼ = 0 , n ≤ − 1 . Lo oking up T able 6, one ﬁnds that these are precisely the v alues rep or ted. 8.2. The group [3 , 4 1 , 1 ] . The Coxeter diagra m fo r this gro up Γ can be found in Figure 2, from whic h the following presentation can be rea d oﬀ (see Sectio n 2): h w, x, y , z | w 2 = x 2 = y 2 = z 2 = 1 , ( wx ) 3 = ( xy ) 4 = ( y z ) 2 = ( z w ) 2 = ( wy ) 2 = ( xz ) 4 = 1 i . This g roup acts on H 3 with coﬁnite volume, with fundamental do ma in a (non- compact) 3-simplex ∆ 3 with one idea l vertex. After lab eling the hyper pla nes ex- tending the four faces by the four genera tors of Γ, the angles b etw een these hyper- planes s atisfy the following r elationships (see Section 2): 30 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ • ∠ ( P w , P y ) = ∠ ( P w , P z ) = ∠ ( P w , P z ) = π / 2, • ∠ ( P w , P x ) = π / 3, • ∠ ( P x , P z ) = ∠ ( P x , P y ) = π / 4. The ideal vertex arises as the intersection (at inﬁnity) of the three hyperplanes P x ∩ P y ∩ P z , and has stabilizer the 2-dimensio nal crystallo graphic gro up [4 , 4]. While the action of Γ on H 3 do es not give a co compact mo del for E F I N (Γ), one can o btain such a mo del by Γ-equiv ar iantly truncating disjoint horospher es centered at the Γ-or bits of the idea l vertex (see Section 7). The splitting formula (see Cor ollary 3.4) tells us that w e hav e, for a ll n ≤ 1, isomorphisms: K n ( Z Γ) ∼ = H Γ n ( E F I N (Γ); K Z −∞ ) ⊕ k M i =1 H V i n ( E F I N ( V i ) → ∗ ) . Let us now ident ify the (ﬁnitely many) g roups { V i } that a pp ear in the ab ov e formula. As explained in Section 4, these gro ups will ar ise as stabilizers of T yp e I geo desics, which are precisely (up to the Γ-a ction) one of the six geo desics P w ∩ P x , P w ∩ P y , P w ∩ P z , P x ∩ P y , P x ∩ P z , and P y ∩ P z . Note that since the geo desic segments P x ∩ P y , P y ∩ P z and P x ∩ P z pro ject to non-co mpact segments in the fundamental domain (they give rise to edges joined to the ideal vertex), these geo desics will never have an inﬁnite stabilizer, a nd we can hence sa fely ignor e them. T o identify the stabilizer s of the remaining three geo des ics, we follow the pro - cedure fr o m Sec tion 4. W e ﬁrst need to iden tify the (non-ideal) vertex stabilizers for the simplex ∆ 3 . Recall that these will b e the sp ecial subgr oups generated by triples of genera tors. But from the C oxeter dia gram for Γ, one immedia tely se e s that the triple of vertices span out the sub diagr ams: • the Coxeter g roup [3 , 4] ∼ = S 4 × Z / 2 will b e the stabilizer of the vertices P w ∩ P x ∩ P z and of the vertex P w ∩ P x ∩ P y , • the gr oup ( Z / 2) 3 will be the stabilizer o f the vertex P w ∩ P y ∩ P z . Now for each of the three (p otentially co co mpact) type I geo desics that we have ( P w ∩ P x , P w ∩ P y , and P w ∩ P z ) one can consider the pro jection to the fundament al domain ∆ 3 . F rom T able 1, lo oking up the v ertex sta bilizers S 4 × Z / 2, w e see that every o ne of the three geo desics pro jects to precisely the asso ciated edge in ∆ 3 . T o ﬁnd the stabilizers o f these geo desics, we now us e Bass- Serre theory as ex- plained in Se c tio n 4. T o ﬁnd the vertex g roups, one uses T a ble 1, while the edge group will b e pr ecisely the dihedra l group given by the sp ecial subgroup asso ciated to the geo desic. This immediately g ives us the stabilizers: • one cop y of D 6 ∗ D 3 D 6 , corresp onding to the g eo desic P w ∩ P x , • t wo c opies of ( Z / 2 × D 2 ) ∗ D 2 ( Z / 2 × D 2 ) ∼ = D 2 × D ∞ , cor resp onding to the t wo geo desics P w ∩ P y and P w ∩ P z , which ar e precisely the gr oups rep orted in T a ble 3. Finally , a mongst these three subgroups, one needs to decide which o nes hav e a non- trivial co kernel for the rela tiv e assembly map. Thes e cokernels are listed out in T able 6, and one sees that the only non-trivial contribution will come from the tw o co pies of D 2 × D ∞ , each of which will contribute L ∞ Z / 2 to the ˜ K 0 ( Z Γ) and W h (Γ). So w e are ﬁnally left with computing the homology coming from the ﬁnite sub- groups, i.e. the term H Γ n ( E F I N (Γ); K Z −∞ ). As we men tioned ea r lier, a co compact fundamen tal domain for H 3 / Γ is g iven by “ truncating” the idea l vertex from ∆ 3 . LOWER ALGEBRAIC K -THEOR Y OF HYPERBOLIC 3-SIMPLE X REFLECTION GR OUPS. 31 The stabilizers o f cells in the fundamen tal domain can b e r ead oﬀ fr o m the Cox- eter dia gram, as they will precisely b e the sp ecial subgroups (see the discussion in Section 7). W e see that: • there is one 3-dimensional cell (the interior o f ∆ 3 ), with trivial stabilizer, • there are ﬁve 2-dimensional cells, with stabilizer Z / 2 (for the face s of the original ∆ 3 ), or triv ia l (for the face coming from trunca ting the ideal vertex in ∆ 3 ), • there are nine 1-dimensional cells (the six e dg es of ∆ 3 , and thre e edges obtained fro m the truncation). Three of these will have stabilizer Z / 2 (those coming from truncating the ideal vertex in ∆ 3 ), three will have stabilizer D 2 (from the edges corr esp onding to P y ∩ P z , P w ∩ P y , and P w ∩ P z ), tw o will hav e stabilizer D 4 (from the edges corres p onding to P x ∩ P y and P x ∩ P z ), and one with stabilizer D 3 (from the edge cor resp onding to P w ∩ P x ), • there are six 0-dimensional cells (three non-idea l vertices o f ∆ 3 , and three from the truncatio n of the ideal vertex). Two hav e stabilizer s D 4 (from the truncation of the tw o edg es with the same stabilizer), one has stabilizer D 2 (from the truncation of the third edge), tw o hav e stabilizer S 4 × Z / 2 (from t wo of the non- ideal vertices), and o ne has stabilizer ( Z / 2) 3 (from the third non-ideal v ertex). Now to obtain the E 2 -terms in the Q uinn sp ectral seq ue nc e , we need the homology of the complex: · · · → M σ p +1 W h q (Γ σ p +1 ) → M σ p W h q (Γ σ p ) → M σ p − 1 W h q (Γ σ p − 1 ) · · · → M σ 0 W h q (Γ σ 0 ) , where σ p are the p - dimensional cells (which we identiﬁed ab ov e). But fro m the work in Section 5, we know explicitly all the groups app ear ing in the a bove complex. Indeed, lo oking up the non-zero K -gro ups in T able 5, we se e tha t the only one of t he c el l stabilizers that has non-trivial K -the ory is the gr oup S 4 × Z / 2. There are tw o copies o f this g r oup, arising as stabilizers of 0-cells, and w e hav e that K − 1 ( Z [ S 4 × Z / 2]) ∼ = Z and ˜ K 0 ( Z [ S 4 × Z / 2]) ∼ = Z / 4. This immediately tells us that non-zer o terms in the Quinn sp ectral seque nc e will b e E 2 0 , − 1 ∼ = Z 2 and E 2 0 , 0 ∼ = ( Z / 4) 2 . This implies that the sp ectra l seq ue nc e immediately co llapses, giving us that H Γ n ( E F I N (Γ); K Z −∞ ) ∼ = 0 for n < − 1 , n = 1, and H Γ − 1 ( E F I N (Γ); K Z −∞ ) ∼ = Z 2 , H Γ 0 ( E F I N (Γ); K Z −∞ ) ∼ = ( Z / 4) 2 . W e now have b oth the terms app ear ing in the splitting formula, and we conclude that the low er algebr a ic K -theory o f the gro up Γ is given by: W h n (Γ) =          W h (Γ) ∼ = L ∞ Z / 2 , n = 1 ˜ K 0 ( Z Γ) ∼ = ( Z / 4) 2 ⊕ L ∞ Z / 2 , n = 0 K − 1 ( Z Γ) ∼ = Z 2 , n = − 1 K n ( Z Γ) ∼ = 0 , n ≤ − 1 . Lo oking up T able 7, one ﬁnds that these are precisely the v alues rep or ted. 32 JEAN-FRANC ¸ OIS LAFONT AND IVONNE J. OR TIZ References [Bar03] A. Bartels. On the domain of the assembly map in algebr aic K -the ory. Algebr. Geom. T op ol. 3 (2003), 1037-1050. [Bas65] H. Bass. The Dirichlet unit the or em, induc e d char acters, and White he ad gr oups of ﬁnite g r oups. T op ology 4 (1965), 391-410. [Bas68] H. Bass. Algebr aic K -the ory. 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Algebraic K-theory of hyperbolic 3-simplex reflection groups

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