A central limit theorem for connected components of random coverings of manifolds with nilpotent fundamental groups

A CENTRAL LIMIT THEOREM F OR CONNECTED COMPONENTS OF RANDOM CO VERINGS OF MANIF OLDS WITH NILPOTENT FUND AMENT AL GR OUPS ABDELMALEK ABDESSELAM Abstract. There is a w ell understo o d w ay of generating random cov erings of a fixed man- ifold b y sampling homomorphisms from the fundamental group of this manifold in to the symmetric group. W e pro ve a cen tral limit theorem for the n um b er of connected compo- nen ts of these random cov erings when the fundamental group is nilp oten t. This provides a nonab elian ge neralization of an earlier result by the author and Shannon Starr in the case of the torus where the fundamental group is a free abelian group of rank at least tw o. Our result relies on the work of du Sautoy and Grunewald on the subgroup gro wth zeta func- tions of nilp otent groups, and on Delange’s generalization of the Wiene r-Ik ehara T aub erian theorem. Contents 1. In tro duction 1 2. Preliminaries 5 2.1. T op ology 5 2.2. Com binatorics 7 2.3. Group theory 8 2.4. T aub erian theory 10 3. Setting up the saddle p oint analysis 14 4. Ma jor and minor arc estimates 16 5. Completion of the pro of of the main theorem 19 6. Outlo ok 20 References 20 1. Intr oduction Let X b e a fixed nonempty top ological space. As is customary in the theory of the fundamen tal group and co v ering spaces, w e will assume that X is path-connected, lo cally path-connected, and semilo cally simply connected. W e pic k a basep oint x 0 ∈ X and denote the fundamen tal group based at x 0 b y G = π 1 ( X , x 0 ). There is a w ell kno wn correspondence, in fact equiv alence of categories, b etw een top ological cov erings π : Y → X and left actions L : G × E → E of the group G on some set E (see [23, pp. 68–70] and [41, Thm. 2.3.4]). The precise corresp ondence as well as other useful definitions will b e recalled in § 2, for the b enefit of the reader who may not b e an exp ert in all mathematical areas relev an t to the presen t article: probabilit y theory , group theory , top ology , and num b er theory . W e will only consider finite co v ers, where ( Y , π ) is n -sheeted, with n some nonnegativ e in teger. The 1 corresp onding set E is finite with | E | = n , where w e use the notation | · | for the cardinality of finite sets. The num b er of connected comp onen ts of the co ver will b e denoted by c ( Y , π ) and it is equal to the n umber of orbits for the L action of G on E whic h we will similarly denote by c ( E , L ). Suc h an action is the same as a group homomorphism G → S E in to the group S E of bijections E → E which is the symmetric group S n if E = [ n ] := { 1 , 2 , . . . , n } . W e no w add the hypothesis that G is finitely generated, for instance if one requires X to b e Hausdorff compact. Then the set of homomorphisms Hom( G, S n ) is finite. By randomly sampling an element φ ∈ Hom( G, S n ) w e immediately get a notion of random cov er ( Y , π ) for X , as w ell as a notion of random manifold Y . This kind of mo del for random manifolds has receiv ed muc h recent attention, e.g., in the articles [8, 29, 30, 31, 32]. Although concerned with enumeration of cov erings rather than probabilistic mo dels, similar constructions were used in [13, 19] (for ramified cov ers where X is a space minus the ramification lo cus). F or other related work on the en umeration of co vers, see also [27] and the review [26]. A natural problem to consider in this context, is the asymptotic study when n → ∞ of the random v ariables giv en by the Betti n umbers of these random manifolds given as co v ers of a fixed space X . Results of this kind are v ery sensitive to the nature of the group G , and in particular its subgroup growth prop erties [28]. F or instance, [8] establish such results for groups which are central extensions of free pro ducts of finite cyclic groups. In this article, w e establish a cen tral limit theorem (CL T) for the zero-th Betti num ber c ( Y , π ) of these random manifolds, under the hypothesis that G is close to commutativ e, i.e., is nilp otent together with some mild technical conditions. Our main theorem generalizes the one of [6] whic h corresp onds to the ℓ -dimensional torus X = T ℓ = R ℓ / Z ℓ , where G = Z ℓ . This article follo ws the same pro of strategy as in [6], with additional ingredients pro vided by the work of du Sauto y and Grunew ald on subgroup growth for nilpotent groups [15], as w ell as Delange’s generalization of the Wiener-Ikehara T aub erian theorem (see [34]). F or n ≥ 0, and 0 ≤ k ≤ n , we denote by A ( G, n, k ) the cardinalit y of the set of homo- morphisms φ ∈ Hom( G, S n ) for whic h c ([ n ] , L ) = k , where L is the left-action given by φ , namely , L : G × [ n ] → [ n ], ( g , i ) 7→ L ( g , i ) = σ ( i ) with the p ermutation σ = φ ( g ). With another sligh t abuse of notation, w e will denote c ( φ ) := c ([ n ] , L ). In terms of a new v ariable x , w e define the p olynomial H G,n ( x ) = 1 n ! n X k =0 A ( G, n, k ) x k . When x > 0 is a fixed p ositiv e real n um b er, w e can define the probability mass function P G,n,x on the set Hom( G, S n ) b y P G,n,x ( φ ) = x c ( φ ) n ! H G,n ( x ) . While the abov e-men tioned work on random co v ers t ypically considers uniform sampling, i.e., x = 1, w e study the more general biased measures P G,n,x whic h are anlogues of the Ew ens measure on random p ermutations (the case of the circle X = S 1 with G = Z ). When the homomorphism φ is sampled according to P G,n,x , this giv es rise to the random v ariable K G,n := c ( φ ) = c ([ n ] , L ) = c ( Y , π ). This is the random num b er of connected comp onents or orbits for whic h w e prov e a CL T. In order to state the latter, we need to quickly recall some 2 definitions and results from the theory of subgroup gro wth [28], with more details provided in § 2. F or a group G , and for in tegers n ≥ 1, we let a n ( G ) := { H ≤ G | [ G : H ] = n } which coun ts the n um b er of subgroups (not necessarily normal) of index exactly n . When G is finitely generated, w e hav e a n ( G ) < ∞ , for all n . The group G is called a T -group, when it is finitely generated, torsion-free, and nilpotent. In this case, the a n ( G ) gro w at most p olynomially in n , and this allows one to define the Dirichlet series, in tro duced in [22], ζ G ( s ) := ∞ X n =1 a n ( G ) n s , the subgroup growth zeta function of the group G whic h generalizes the Riemann zeta function corresp onding to G = Z . This Diric hlet series conv erges in the half-plane Re( s ) > α G ≥ 0, where α G := lim sup n →∞ ln  P n j =1 a j ( G )  ln n . When α G > 0, this quantit y is the same as the abscissa of con v ergence of the Diric hlet series, i.e., the infim um of the set of real num bers σ such that the series con verges for Re( s ) > σ . Since the coefficients a n ( G ) are ob viously nonnegative, this also coincides with the abscissa of absolute conv ergence, whic h is defined similarly with the con vergence requirement replaced b y that of absolute conv ergence. Since this do es not seem to b e a standard definition, let us sa y that a finitely generated group has at least linear subgroup growth if ∃ c > 0 , ∀ n ≥ 1 , a n ( G ) ≥ cn . F or such a group, we must ha ve α G ≥ 2. The follo wing is a deep result by du Sautoy and Grunew ald [15]. Theorem 1.1. L et G b e an infinite T -gr oup, then α G is a r ational numb er, and ther e exists δ > 0 such that ζ G ( s ) admits a mer omorphic c ontinuation to the domain Re( s ) > α G − δ . Mor e over, ζ G ( s ) has a p ole at s = α G , and no other singularity on the line Re( s ) = α G . Assuming the conclusions of the theorem hold, we let m G ≥ 1 denote the order of the p ole at α G , and let γ G denote the co efficient of the most singular term 1 ( s − α G ) m G for the Laurent expansion of ζ G ( s ) at α G . Clearly γ G = lim s → α G ( s − α G ) m G ζ G ( s ) > 0. W e can now define the constan t K G = Γ( α G ) γ G ( m G − 1)! , where Γ( s ) is the Euler gamma function. W e can no w state our main theorem. Theorem 1.2. Supp ose G is a T -gr oup with at le ast line ar sub gr oup gr owth, then the le ading asymptotics of the me an and varianc e of the r andom variables K G,n ar e given by E K G,n ∼ 1 α G − 1 × α G −  m G − 1 α G  × ( x K G ) 1 α G × n α G − 1 α G × (ln n ) m G − 1 α G , V ar( K G,n ) ∼ 1 α G ( α G − 1) × α G −  m G − 1 α G  × ( x K G ) 1 α G × n α G − 1 α G × (ln n ) m G − 1 α G . 3 Mor e over, the normalize d r andom variables K G,n − E K G,n p V ar( K G,n ) c onver ge in distribution and in the sense of moments to the standar d Gaussian N (0 , 1) . Namely, we have lim n →∞ E " f K G,n − E K G,n p V ar( K G,n ) !# = 1 √ 2 π ∞ Z −∞ f ( s ) e − s 2 2 d s , for al l f ’s that ar e b ounde d c ontinuous functions, or p olynomials. The following corollary giv es a more practical v ersion of the theorem with a sufficien t condition whic h is easier to chec k, and which inv olv es the notion of Hirsch length (see § 2 for a refresher). F or a p olycyclic group G , we will denote by h ( G ) the Hirsc h length of G . W e will also denote by G ab the ab elianization of G . Corollary 1.1. The c onclusion of The or em 1.2 holds if G is a T -gr oup with h ( G ab ) ≥ 2 . Useful running examples to help understand our result are giv en by the torus as well as the Heisen b erg manifold. First consider the torus X = T ℓ with G = Z ℓ with the restriction to ℓ ≥ 2. As a consequence of results by Hermite [25, p. 193] and Eisenstein [18, p. 355] (see also the earlier [17, p. 330] for ℓ = 2), the zeta function is then given by ζ Z ℓ ( s ) = ζ ( s ) ζ ( s − 1) · · · ζ ( s − ℓ + 1) , in terms of the Riemann zeta function. See also [28, Ch. 15] for fiv e different pro ofs of this b eautiful classical result. The abscissa of con vergence is α Z ℓ = ℓ , and the corresp onding p ole is simple, i.e., m Z ℓ = 1. Then γ Z ℓ = ζ (2) ζ (3) · · · ζ ( ℓ ) and K Z ℓ = ( ℓ − 1)! ζ (2) ζ (3) · · · ζ ( ℓ ) whic h is the same as the constant denoted K ℓ in [6]. In this case, our new theorem agrees with [6, Thm 1.1]. Note that the CL T also holds for ℓ = 1, but requires differen t proof tec hniques, and this is wh y w e restrict to ℓ ≥ 2. F or ℓ = 1, x = 1 this is the classic theorem of Gonc haro v for the n umber of cycles of a uniformly random p erm utation, while for ℓ = 1, x  = 1 this is a result by Hansen (see [6] and references therein) with sampling according to the Ew ens measure. A more interesting nonab elian example is giv en b y the Heisenberg manifold. Let Heis( R ) b e the group, under matrix multiplication, of matrices of the form   1 x z 0 1 y 0 0 1   where x, y , z are arbitrary real num bers. Let Heis( Z ) b e the same matrix group but with x, y , z in Z instead. The Heisenberg manifold is the quotient X = Heis( R ) / Heis( Z ). See[36, § 7] for a v ery efficient in tro duction to the basic top ological prop erties of X , e.g., compactness, and the structure as a circle bundle ov er the t wo-dimensional torus. The fundamen tal group is just G = Heis( Z ) whic h is nilp otent of step 2. W e also ha ve h (Heis( Z )) = 3 while h (Heis( Z ) ab ) = 2. By a result of Smith [38, Thm. 3, p. 20], the corresp onding zeta function is ζ Heis( Z ) ( s ) = ζ ( s ) ζ ( s − 1) ζ (2 s − 2) ζ (2 s − 3) ζ (3 s − 3) . (1) 4 Note that zeta functions ζ G whic h are as explicit are hard to come by . F or a thorough study through sp ecific examples see [16], and also the review [43] for an introduction to this area. Another useful, and more recent, relev ant reference in the area is [40]. F rom the ab ov e explicit formula, w e see that α Heis( Z ) = 2 where the zeta function has a double p ole, i.e., γ Heis( Z ) = 2. Moreov er, near s = 2, w e hav e ζ ( s ) Heis( Z ) ∼ ζ (2) 2 2 ζ (3)( s − 2) 2 , hence γ Heis( Z ) = ζ (2) 2 2 ζ (3) . Our main theorem applies to G = Heis( Z ) and sho ws that K Heis( Z ) ,n satisfies the CL T with mean and v ariance asymptotics given by E K Heis( Z ) ,n ∼ π 2 12 p ζ (3) × √ x × √ n ln n , and V ar( K Heis( Z ) ,n ) ∼ π 2 24 p ζ (3) × √ x × √ n ln n . 2. Preliminaries Giv en the m ultidisciplinary asp ect of this article, we collect in this section basic material ab out the notions inv olv ed in this work. W e cannot prov e all statements and refer instead to the relev an t literature, but we will try to pro vide all the definitions. 2.1. T op ology. Our main reference for this subsection is [23, Ch. 1]. A path in space X is a con tin uous map α : [0 , 1] → X . One has a notion of concatenation αβ of t wo paths α , β for whic h α (1) = β (0). The concatenation αβ is obtained b y going o v er α first, and then going o v er β as a second step. W e denote b y [ α ] the homotopy class of the pat h α corresp onding to con tin uous deformation with fixed endp oin ts. The fundamen tal group G = π 1 ( X , x 0 ) is the set of classes [ α ] for paths whic h start and end at x 0 , with the m ultiplication [ α ][ β ] := [ αβ ]. The reverse path α is defined b y α ( t ) := α (1 − t ) and gives the inv erse op eration [ α ] − 1 = [ α ]. The neutral element is [ x 0 ], the class of the constan t path equal to x 0 . Finite cov erings π : Y → X of the fixed space X form a category denoted by Cov . The ob jects are finite co verings ( Y , π ). A morphism f from ( Y 1 , π 1 ) to ( Y 2 , π 2 ) is a contin uous map f : Y 1 → Y 2 for which π 2 ◦ f = π 1 . The categorical definition of an isomorphism of co v erings is therefore such a map f whic h is a homeomorphism. If ( Y 1 , π 1 ) = ( Y 2 , π 2 ), suc h an isomorphism f is called an automorphism of that co ver. W e will denote b y Aut( Y , π ) the automorphism group, under comp osition, of the cov er ( Y , π ). W e stress that w e allow co v erings where Y is disconnected. W e even allow empt y cov erings where Y = ∅ and π is the map with empty graph. W e now also in tro duce the category Act of finite sets with G -action. An ob ject is a pair ( E , L ) where E is a finite set and L : G × E → E is a left G -action on the set E . A morphism f from ( E 1 , L 1 ) to ( E 2 , L 2 ) is a map f : E 1 → E 2 suc h that f ( L 1 ( g , a )) = L 2 ( g , f ( a )) for all g ∈ G and a ∈ E 1 . Namely , suc h morphisms are G -equiv arian t maps. One again has a notion of isomorphism and automorphism. The group of automorphisms of an ob ject ( E , L ) 5 will b e denoted b y Aut( E , L ). Note that w e ma y switch p ersp ective by identifying a left action L on E with the asso ciated group homomorphism φ ∈ Hom( G, S E ) defined in § 1. F or a fixed basep oint x 0 , we no w introduce tw o functors relating these tw o categories: a functor L from the category of finite co vers of X to that of finite sets with left G -action, and a functor C going in the opp osite direction. Consider an ob ject ( Y , π ) in the former category and define E = π − 1 ( { x 0 } ), the fib er o ver x 0 . Let [ α ] ∈ G , a ∈ E , and let e α denote the lift starting at a of the path α in X . Namely , e α is the path in Y such that π ◦ e α = α , and e α (0) = a . By the path lifting lemma, this lift exists and is unique. W e let b = e α (1) ∈ E whic h is indep endent of the choice of represen tativ e α , by the homotop y lifting lemma. W e then define L ([ α ] , a ) := b which is a left G -action on the fib er E . This construction is called the p ermutation represen tation in [23, pp. 68–70], and the mono drom y action on the fib er in [41, Ch. 2]. Note that if one did not rev erse the path, w e w ould ha v e obtained a righ t instead of left action. This is an unpleasant feature due to the ab ov e standard definition of m ultiplication in the fundamental group. This resonates with [41, Remark 2.3.1] which advocates for the opp osite conv en tion used by Deligne. The probabilist reader will recognize the very similar issue in the theory of Marko v c hains in finite state spaces, and the dilemma of chosing to represent probability distributions b y row or column vectors. At the lev el of ob jects, the functor L send ( Y , π ) to L ( Y , π ) = ( E , L ) just constructed. No w consider a morphism f : ( Y 1 , π 1 ) → ( Y 2 , π 2 ). The restriction (and co-restriction) to the fib er f | E 1 : E 1 → E 2 , where for i = 1 , 2, E i = π − 1 i ( { x 0 } ). This map is G -equiv arian t is thus is a morphism ( E 1 , L 1 ) → ( E 2 , L 2 ) where L i is defined as ab ov e. Hence the functor sends f to the morphism L [ f ] := f | E 1 . In order to giv e the definition of the functor C , w e first recall the definition of the univ ersal co v er whose construction requires the prop ert y of semilo cal simple connectedness mentioned at the b eginning of § 1. The universal cov er b π : b X → X is a co ver such that b X is simply connected and it is essentially unique. One wa y to view it concretely is to see it as the set of homotop y classes of [ ρ ] of paths ρ in X with ρ (0) = x 0 but ρ (1) unrestricted. The pro jection is simply b π ([ ρ ]) := ρ (1) consisting in only remembering the final destination of the path. F or [ α ] ∈ G and [ ρ ] ∈ b X , it is easy to see that ([ α ] , [ ρ ]) 7→ [ α ][ ρ ] := [ α ρ ] giv es a left action of G on b X . F or fixed [ α ], the map [ ρ ] 7→ [ α ][ ρ ] b elongs to Aut( b X , b π ). This action is transitive when restricted to fib ers, i.e., if [ ρ 1 ] and [ ρ 2 ] pro ject to the same endp oin t in X , there exists an element [ α ] ∈ G such that [ α ][ ρ 1 ] = [ ρ 2 ]. This also is a free action, i.e., if [ α ] ∈ G and [ ρ ] ∈ b X are suc h that [ α ][ ρ ] = [ ρ ], then we m ust ha ve [ α ] = [ x 0 ]. W e can now define the functor C . Giv en a finite set E with left-action L by G , we define Y := ( b X × E ) /G the space of orbits for the diagonal action of G on the Cartesian pro duct of the univ ersal co ver with the set E . Namely , w e consider pairs ([ ρ 1 ] , a 1 ) , ([ ρ 2 ] , a 2 ) ∈ b X × E to b e equiv alen t if there exists [ α ] ∈ G , such that [ α ][ ρ 1 ] = [ ρ 2 ] and L ([ α ] , a 1 ) = a 2 . W e denote b y [([ ρ ] , a )] the equiv alence class of a pair ([ ρ ] , a ). W e let π : Y → X b e defined by π ([([ ρ ] , a )]) := b π ([ ρ ]) = ρ (1). The co vering ( Y , π ) =: C ( E , L ) is the ob ject in Cov given by the functor C . Now let ( E 1 , L 1 ), ( E 2 , L 2 ) b e t wo ob jects in Act and let ( Y 1 , π 1 ), ( Y 2 , π 2 ) b e the corresponding ob jects obtained b y C , as just explained. If ϕ : Y 1 → Y 2 is a G -equiv arian t map, then f : Y 1 → Y 2 giv en b y [([ ρ ] , a )] 7→ [([ ρ ] , ϕ ( a ))] is well defined and is a morphism of co v erings of X . W e th us let, C [ ϕ ] := f . 6 Giv en a cov ering ( Y , π ), w e define a map ω Y ,π : ( b X × E ) /G → Y , where E is the fib er π − 1 ( { x 0 } ). F or [ ρ ] ∈ b X and a ∈ E , w e let e ρ b e the lift to Y of the path ρ in X , with the initial condition e ρ (0) = a . By definition, we let ω Y ,π ([([ ρ ] , a )]) := e ρ (1). It is not hard to see that this is an isomorphism of co vers. It is also not hard to see that for all morphisms of co v ers f : ( Y 1 , π 1 ) → ( Y 2 , π 2 ) w e ha v e the comm utativ e diagram prop erty f ◦ ω Y 1 ,π 1 = ω Y 2 ,π 2 ◦ C [ L [ f ]] . In other words, the ω ’s provide a natural isomorphism b et ween C ◦ L and the iden tity functor of the category Cov . Giv en a finite set E with left action L , we define a map λ E ,L : E → π − 1 ( { x 0 } ), where π is the pro jection map of the co v er Y = ( b X × E ) /G obtained as C ( E , L ). By definition, w e let λ ( a ) := [([ x 0 ] , a )], using the constant path equal to the basep oin t x 0 . It is not hard to see that this is an isomorphism of finite sets with left G -action. Moreov er, if ϕ : ( E 1 , L 1 ) → ( E 2 , L 2 ) is a G -equiv arian t map, we hav e the commutativ e diagram prop erty L [ C [ ϕ ]] ◦ λ E 1 ,L 1 = λ E 2 ,L 2 ◦ ϕ . Hence, the λ ’s provide a natural isomorphism b etw een the identit y functor of the category Act and the comp osition functor L ◦ C . The previous discussion can b e summarize b y the follo wing theorem. Theorem 2.1. The natur al tr ansformations ω , λ ar e isomorphisms of functors, and they establish an e quivalenc e of c ate gories b etwe en Cov , the c ate gory of finite c overings of X , and Act , the c ate gory of finite sets with a left gr oup action by G . 2.2. Com binatorics. F or a cov er ( Y , π ) as in the previous section, w e denote by deg( Y , π ) := | π − 1 ( { x 0 } ) | the n umber of sheets or degree of the co v er. In the ring of formal p o wer series C [[ x, z ]] we define the biv ariate generating series G G ( x, z ) := X [ Y ,π ] x c ( Y ,π ) z deg( Y ,π ) | Aut( Y , π ) | , whic h keeps track of the num ber of connected comp onen ts c ( Y , π ) and the n umber of sheets deg( Y , π ) of a co v er ( Y , π ). The summation is o ver all isomorphism classes [ Y , π ] of co vers ( Y , π ). Clearly , they can b e collected in a coun table set. The need for normalizing by the symmetry factor | Aut( Y , π ) | is w ell known in ph ysics and the com binatorics of F eynman diagrams (see [13, 19]), in relation to Joy al’s theory of combinatorial sp ecies (see [1] or the up coming [4]). A consequence of the equiv alence of categories in Theorem 2.1 is that we can rewrite the same generating function as G G ( x, z ) = X [ E ,L ] x c ( E ,L ) z | E | | Aut( E , L ) | , (2) where w e sum ov er isomorphism classes [ E , L ] of pairs ( E , L ) made of a finite set E and a left G -action L on E . This notion, in addition to a category , gives rise to a combinatorial sp ecies, i.e., a functor F from the category of finite sets with bijections into itself (see, e.g., [1]). T o an ob ject E , the functor asso ciates the set F ( E ) of all left actions L b y G on E . F or a morphism, i.e., bijection σ : E 1 → E 2 , the functor asso ciates the relabeling bijection F [ σ ] : F ( E 1 ) → F ( E 2 ) which sends L 1 on E 1 to the action L 2 ( g , a ) := σ ( L 1 ( g , σ − 1 ( a ))) 7 on E 2 . The series (2) is an example of exp onential generating series for the combinatorial sp ecies F of left G -actions. W e can also write G G ( x, z ) = ∞ X n =0 X L ∈ F ([ n ]) x c ([ n ] ,L ) z n n ! , as a consequence of the orbit-stabilizer theorem, or b y [1, Theorem 8] for the natural trans- formation from F to the trivial combinatorial sp ecies. Via the identification of left actions L ∈ F ( E ) with group homomorphisms φ ∈ Hom( G, S E ) men tioned in § 1, w e can rephrase the previous equation as G G ( x, z ) = ∞ X n =0 X φ ∈ Hom( G, S n ) x c ( φ ) z n n ! = ∞ X n =0 H G,n ( x ) z n . By a very general form ula b y Banta y [9, Eq. (1)], based on the so-called exp onential formula, w e ha ve G G ( x, z ) = exp X H ≤ G x z [ G : H ] [ G : H ] ! . This connects the top ological en umeration of cov ers to the subgroup growth function, since w e can also write G G ( x, z ) = exp x ∞ X n =1 a n ( G ) z n n ! . (3) W e refer to [4] for more details, and in particular the deriv ation of the Bryan-F ulman formula, whic h is the sp ecial case G = Z ℓ (see [11] and [5]). 2.3. Group theory. Our main reference for the group theory inv olv ed will b e [14, § 6.1] for the elemen tary notions, as w ell as [37] for the more adv anced treatmen t of p olycyclic groups, and [28] for that of subgroup growth. F or a general group G , w e define the commutator [ x, y ] of tw o elemen ts x, y ∈ G by [ x, y ] := x − 1 y − 1 xy . F or arbitrary subsets A, B ⊆ G , we denote by [ A, B ] the subgroup of G generated by the subset { [ x, y ] | ( x, y ) ∈ A × B } . It is easy to chec k that H ⊴ G , i.e., H is a normal subgroup of G if and only if [ G, H ] ⊆ H . Moreo v er, in that case, G/H is an ab elian group if and only if [ G, G ] ⊆ H . By these tw o remarks and an easy induction, the low er cen tral series (of subgroups) defined by G 0 = G , and G i +1 := [ G, G i ] for all i ≥ 0, is suc h that G = G 0 ⊵ G 1 ⊵ G 2 ⊵ G 3 ⊵ · · · , and G i /G i +1 is ab elian for all i ≥ 0. F or i = 0, this is the ab elianization G ab := G/ [ G, G ]. If the series reac hes the trivial group { 1 G } in finitely man y steps, then G is called a nilp oten t group. If n is the smallest i for whic h G i = { 1 G } , then we sa y that G is n -step nilp oten t. Nilp oten t of step zero means the group is trivial. Nilp otent of step one means the group is ab elian. Finally , an interesting example of nilp oten t group of step t wo is G = Heis( Z ). Indeed, consider the matrices X =   1 1 0 0 1 0 0 0 1   , Y =   1 0 0 0 1 1 0 0 1   , Z =   1 0 1 0 1 0 0 0 1   , 8 whic h are easily seen to generate the Heisen b erg group. A quic k computation sho ws [ X , Y ] = Z , [ X , Z ] = I , [ Y , Z ] = I . Hence G 0 = G , G 1 = [ G, G ] = ⟨ Z ⟩ (the infinite cyclic group generated b y Z ), and finally G 2 = { I } , which explicitly v erifies the prop ert y of G b eing 2-step nilp otent. W e also ha ve G ab ≃ Z 2 . No w let G b e a group, and assume that there is a finite sequence of subgroups G = H 0 ⊵ H 1 ⊵ H 2 ⊵ · · · ⊵ H n = { 1 G } , whic h terminates with the trivial group and such that, H i /H i +1 is a cyclic group for all i , 0 ≤ i ≤ n − 1. Then, G is called a p olycyclic group. The num ber h ( G ) of v alues of i for whic h H i /H i +1 is infinite, i.e., H i /H i +1 ≃ Z is an inv arian t of the group G . Namely , it is indep enden t of the choice of sequence ( H i ). It is called the Hirsch length of G . If G is a finitely generated ab elian group, then b y the fundamental structure theorem for such groups, G ≃ Z r ⊕ T orsion, where T orsion is a finite direct sum of finite cyclic groups. It is easy to see that, in this ab elian situation, the Hirsc h length h ( G ) coincides with the torsion-free rank r . One thus easily chec ks, for the Heisenberg group that h (Heis( Z )) = 3, and h (Heis( Z ab ) = 2, as mentioned in § 1. The follo wing lemma is easy to prov e, and giv es con v enien t wa ys to c hec k the applicabilit y of Corollary 1.1, and thus our main theorem 1.2. Lemma 2.1. F or a p olycyclic gr oup G , the fol lowing ar e e quivalent: (1) h ( G ab ) ≥ 2 , (2) ∃ ℓ ≥ 2 , ∃ N ⊴ G such that G/ N ≃ Z ℓ , (3) ∃ N ⊴ G such that G/ N ≃ Z 2 , (4) ther e exists a surje ctive gr oup homomorphism G → Z 2 . No w let G b e a group, and for n ≥ 1, let T n ( G ) b e the set of transitive left G -actions on [ n ], i.e., the set of L ∈ F ([ n ]) such that c ([ n ] , L ) = 1. Let S n ( G ) b e the set of subgroups of G of index exactly n . Consider the map Φ n : T n ( G ) → S n ( G ) whic h sends an action L to Φ( L ) := { g ∈ G | L ( g , 1) = 1 } , i.e., the stabilizer of the preferred element 1 ∈ [ n ]. The map is surjective, b ecause one can tak e the canonical action of G on G/H by left multiplication, where H ∈ S n ( G ), and then one can transp ort this action to [ n ] via a bijection which sends the left coset H to 1. The map ( σ, L ) 7→ F [ σ ]( L ) gives a left action of S n on T n ( G ). It is not hard to see that Φ( F [ σ ]( L )) = Φ( L ) for ev ery transitive left action L on [ n ], and for all bijection σ : [ n ] → [ n ] whic h satisfies σ (1) = 1. Moreo ver, one can sho w that tw o transitiv e left G -actions L 1 , L 2 on [ n ] will satisfy Φ( L 1 ) = Φ( L 2 ) if and only if L 2 = F [ σ ]( L 1 ) for some p erm utation σ which fixes the element 1. Therefore, the map Φ is exactly ( n − 1)!-to-one, and a n ( G ) = T n ( G ) ( n − 1)! . Finiteness of a n ( G ) immediately follo ws from this, if the group G is finitely generated. F or a normal subgroup N of G , precomp osition b y the surjectiv e pro jection G → G/ N is an injectiv e map from Hom( G/ N , S n ) to Hom( G, S n ). This implies | T n ( G/ N ) | ≤ | T n ( G ) | and therefore a n ( G/ N ) ≤ a n ( G ), for all n ≥ 1. 9 The men tioned results by Hermite and Eisenstein sho w that a n ( Z ℓ ) = X δ 1 ,...,δ ℓ ≥ 1 1 l { δ 1 · · · δ ℓ = n } δ ℓ − 1 1 δ ℓ − 2 2 · · · δ 1 ℓ − 1 δ 0 ℓ , where 1 l {· · · } denotes the indicator function of the enclosed condition. In particular, a n ( Z 2 ) is given by the sum of divisor function σ ( n ) ≥ n . W e ha ve thus established the following lemma. Lemma 2.2. Supp ose the p olycyclic gr oup G satisfies any of the c onditions state d in L emma 2.1, then a n ( G ) ≥ n for al l n , and as a c onse quenc e, the gr oup G is of at le ast line ar sub gr oup gr owth. 2.4. T aub erian theory. W e now assume that G is a T -group with at least linear subgroup gro wth, so that Theorem 1.1 applies with α G ≥ 2. F or an y real n um b er β , define the function (0 , ∞ ) → R W G,β ( u ) = ∞ X δ =1 δ β a δ ( G ) e − δ u . By the term-b y-term differen tiation theorem, it is easy to see that W ′ G,β ( u ) = − W G,β +1 ( u ). These functions are thus C ∞ on (0 , ∞ ). They take p ositive v alues and they go to 0 when u → ∞ . A k ey ingredien t for the present w ork is the following leading asymptotics result for these function near zero. Prop osition 2.1. If β > − α G , we have, as u → 0 + , the asymptotic e quivalenc e W G,β ( u ) ∼ Γ( α G + β ) γ G ( m G − 1)! × u − ( α G + β ) × ( − ln u ) m G − 1 . F or G = Z ℓ these functions W Z ℓ ,β w ere studied in [6] and [3], where they were denoted Z [ ℓ ] ℓ + β ( u ) if β ∈ Z . See [6, Eqs. (12)-(15)] for the leading asymptotics, and [3, Prop. 2.1] for the complete asymptotics to order O ( u ∞ ) using the Mellin approac h [20, 44] (see also [42]). This requires the meromorphic con tinuation of ζ G to the entire complex plane and detailed kno wledge of the p oles. As the example of the Heisenberg group shows, these p oles may b e non trivial Riemann zeros in disguise, due to the denominator of (1). F or general T -groups G , w e need a less quantitativ e to ol given b y Delange’s generalization of the Wiener-Ikehara T aub erian theorem which requires no information whatso ev er, as to ho w the zeta function b eha ves to the left of the vertical line Re( s ) = α G . The same T aub erian theorem will also pro vide us with a proof of the follo wing result (already in [15]) needed for our ma jor arc estimates. Prop osition 2.2. If β > − α G , we have, as the inte ger N go es to ∞ , the asymptotic e quiv- alenc e N X n =1 δ β a δ ( G ) ∼ γ G ( m G − 1)!( α G + β ) × N α G + β × (ln N ) m G − 1 . The T aub erian theorem w e will use to derive Prop ositions 2.1 and 2.2 is the following. 10 Theorem 2.2. L et b ( t ) b e a me asur able function [0 , ∞ ) → R which is b ounde d on finite intervals, and such that G ( s ) := Z ∞ 0 b ( t ) e − st d t c onver ges (in L eb esgue sense, i.e., absolutely) for al l s ∈ C with Re( s ) > 0 . Supp ose also that ther e exists an inte ger m ≥ 1 and ther e exists ε 0 > 0 such that lim t →∞ tb ( t ) Z ε 0 0 u m − 1 e − tu d u = 1 . L et a ( t ) b e a nonde cr e asing function [0 , ∞ ) → R . Supp ose ther e is some α > 0 such that f ( s ) := Z ∞ 0 a ( t ) e − st d t c onver ges (absolutely) for al l s ∈ C with Re( s ) > α . Supp ose that for some c onstant c 0 > 0 , the function F ( s ) := f ( s ) − c 0 G ( s − α ) c an b e r ewritten as F ( s ) = g ( s ) ( s − α ) m − 1 + h ( s ) , wher e g ( s ) is a p olynomial of de gr e e at most m − 2 if m ≥ 2 , or identic al ly zer o if m = 1 , and wher e h ( s ) is a function which is c ontinuous on Re( s ) ≥ α and holomorphic on the domain Re( s ) > α . Then, the pr evious hyp otheses imply the asymptotics a ( t ) = ( c 0 + o (1)) e αt b ( t ) , as t → ∞ . This theorem is [34, Thm 5.1] to which w e refer for a p edagogical pro of. It is a simpler v ersion of a more general theorem due to Delange [12]. In b oth applications of Theorem 2.2 b elow, we will take m = m G the order of the p ole of the subgroup growth zeta function ζ G ( s ), and for that m , w e will tak e b ( t ) := t m − 1 ( m − 1)! . Th us G ( s ) = 1 s m , and ε 0 = 1 w orks for the relev an t h yp othesis in Theorem 2.2. Pro of of Prop osition 2.1: In keeping with the notations of Theorem 2.2, we define a ( t ) := X δ ≥ 1 δ β a δ ( G ) e − δ e − t , whic h conv erges for t ≥ 0 b ecause of the p olynomial growth of a δ ( G ), and is nondecreasing. Using T onelli’s theorem, for σ := Re( s ) > 0, and performing the change of v ariables w = δ e − t , 11 w e ha ve Z ∞ 0   a ( t ) e − st   d t = Z ∞ 0 a ( t ) e − σ t d t = X δ ≥ 1 δ β a δ ( G ) Z ∞ 0 e − δ e − t e − σ t d t = X δ ≥ 1 δ β a δ ( G ) δ − σ Z δ 0 e − w w σ − 1 d w ≤ Γ( σ ) ζ G ( σ − β ) < ∞ , if w e also require σ > α := α G + β , whic h is strictly p ositiv e b y hypothesis. Hence if Re( s ) > α , we can redo the ab o ve calculation without the | · | and obtain f ( s ) := Z ∞ 0 a ( t ) e − st d t = X δ ≥ 1 δ β a δ ( G ) δ − s Z δ 0 e − w w s − 1 d w = Γ( s ) ζ G ( s − β ) − Ψ( s ) , where Ψ( s ) := X δ ≥ 1 δ β a δ ( G ) δ − s Z ∞ δ e − w w s − 1 d w . It is easy to see that Ψ( s ) is an entire analytic function of s ∈ C . W e let c 0 := Γ( α G + β ) γ G , so the function F ( s ) in Theorem 2.2 b ecomes F ( s ) = Γ( s ) ζ G ( s − β ) − Ψ( s ) − Γ( α G + β ) γ G ( s − α G − β ) m G . By Theorem 1.1, the function ζ G ( s − β ) has a p ole of order m = m G ≥ 1 at α = α G + β > 0. Therefore, the function Γ( s ) ζ G ( s − β ) also has a p ole of the same order at the same lo cation, and one can write the relev ant Laurent expansion as Γ( s ) ζ G ( s − β ) = Γ( α G + β ) γ G ( s − α G − β ) m G + m G − 1 X j =1 ν j ( s − α G − β ) j + Φ( s ) , for some Φ( s ) which is con tinuous for Re( s ) ≥ α G + β and holomorphic for Re( s ) > α G + β , and for some constants ν j . The sum ov er j is b y definition empt y or identically zero if m G = 1. As a result, F ( s ) = g ( s ) ( s − α G − β ) m G − 1 + h ( s ) with the p olynomial g ( s ) := m G − 1 X j =1 ν j ( s − α G − β ) m G − 1 − j , and the function h ( s ) := Φ( s ) − Ψ( s ) , 12 whic h fulfills the requirements of Theorem 2.2. The conclusion of latter entails a ( t ) = (Γ( α G + β ) γ G + o (1)) e ( α G + β ) t b ( t ) , as t → ∞ . Let u > 0 b e related to t b y u := e − t , so that W G,β ( u ) = a ( t ), and t → ∞ corresp onds to u → 0 + . The w an ted asymptotics immediately follo w. □ Pro of of Prop osition 2.2: W e will b e rather brief because the steps are similar to the pro vious proof, and also because this example of application of Theorem 2.2 has b een treated in detail in [34] for β = 0, and the shift b y β is a straightforw ard mo dification. W e again use m = m G , α = α G + β , and the same b ( t ) and G ( s ). W e define a ( t ) := X 1 ≤ δ ≤ e t δ β a δ ( G ) . W e then obtain f ( s ) = ζ G ( s − β ) s , when Re( s ) > α G + β > 0, by h yp othesis. W e let c 0 := γ G α G + β . It is easy to see that the hypotheses of Theorem 2.2 hold, from whic h w e deduce the asymp- totics a ( t ) =  γ G α G + β + o (1)  e ( α G + β ) t b ( t ) , as t → ∞ . W e then switch v ariables to u = e t and restrict to in teger v alues u = N , and the w an ted result is established. □ F or our ma jor arc estimates, w e will only need the case β = 1 of Prop osition 2.2. As for Prop osition 2.1, we will need the cases β ∈ {− 1 , 0 , 1 , 2 } explicitly collected b elo w for ease of reference: W G, − 1 ( u ) ∼ K G α G − 1 × u − α G +1 × ( − ln u ) m G − 1 , (4) W G, 0 ( u ) ∼ K G × u − α G × ( − ln u ) m G − 1 , (5) W G, 1 ( u ) ∼ α G K G × u − α G − 1 × ( − ln u ) m G − 1 , (6) W G, 2 ( u ) ∼ α G ( α G + 1) K G × u − α G − 2 × ( − ln u ) m G − 1 , (7) as u → 0 + . The β = 0 function is particularly imp ortant for this work. The function W G, 0 ( u ) is a smo oth decreasing bijection of (0 , ∞ ) on to itself, and w e will need the follo wing lemma whic h giv es the asymptotics of the (comp ositional) inv erse function W − 1 G, 0 ( w ), as w → ∞ . Lemma 2.3. As w → ∞ , we have W − 1 G, 0 ( w ) ∼ K 1 α G G × α −  m G − 1 α G  G × w − 1 α G × (ln w ) m G − 1 α G . The pro of, which starts with (5), is left to the reader, and it is a direct application of the metho d used in [10, § 2.4]. This is the analogue of [6, Lem 2.4], where G = Z ℓ , ℓ ≥ 2, and whic h had no logarithmic corrections. 13 3. Setting up the saddle point anal ysis F rom no w on w e assume the top ological space X satisfies the conditions stated at the b eginning of § 1, and that its fundamental group G = π 1 ( X , x 0 ) satisfies the hypotheses of our main result, i.e., G is a T -group with at least linear subgroup growth. Theorem 1.2 is a consequence of the following k ey prop osition. Prop osition 3.1. Consider the pr evious r andom variables K G,n with distribution determine d by the me asur e P G,n,x . L et ( a n ) b e a se quenc e of r e al numb ers, and let ( b n ) b e a se quenc e of p ositive r e al numb ers such that, as n → ∞ , a n = xW G, − 1  W − 1 G, 0  n x  + o  n α G − 1 2 α G (ln n ) m G − 1 2 α G  , b n = 1 p α G ( α G − 1) × α G −  m G − 1 2 α G  × ( x K G ) 1 2 α G × n α G − 1 2 α G × (ln n ) m G − 1 2 α G × (1 + o (1)) . Then, for al l s ∈ R , we have lim n →∞ ln E  exp  s  K G,n − a n b n  = s 2 2 . Note that, for the b enefit of the reader already aquainted with [6], w e will follo w the same pro of strategy and will use v ery similar notation, e.g., a n , b n for the needed sequences. There should b e no confusion with the subgroup growth sequence a n ( G ) b ecause the latter will alwa ys include the reference to the group G . The real v ariable s is fixed, and should not lead to confusion with the complex argumen t of ζ G , used in § 2.4, and which is no longer needed. Since the group has p olynomial subgroup growth, the series ∞ X δ =1 a δ ( G ) z δ δ con v erges absolutely for | z | < 1. Therefore, the equality (3) holds not only in the sense of formal p o wer series, but also for fixed x > 0, as an equality of analytic functions on the disk | z | < 1. W e can use Cauc hy’s form ula to extract the co efficien t of z n b y H G,n ( x ) = 1 2 iπ I C ( r ) z − n G G ( x, z ) d z z , where C ( r ) denotes the circle of radius r ∈ (0 , 1) around the origin with counterclockwise orien tation. W e will write this radius as r = e − t for a suitably optimized parameter t ∈ (0 , ∞ ). W e will also split H G,n ( x ) as a pro duct of a prefactor times an integral to b e analyzed b y the saddle p oint metho d: H G,n ( x ) = P n ( x, t ) J n ( x, t ) , where P n ( x, t ) := e nt G G ( x, e − t ) = exp ( nt + xW G, − 1 ( t )) , and J n ( x, t ) := Z π − π j n ( x, t, θ ) d θ 2 π , 14 with j n ( x, t, θ ) := e − inθ × G G ( x, e − t + iθ ) G G ( x, e − t ) . F or y , u > 0 and − π < θ < π , we also define q n ( y , u, θ ) := inθ − y X δ ≥ 1 ( e − u + iθ ) δ δ a δ ( G ) − X δ ≥ 1 ( e − u ) δ δ a δ ( G ) ! . (8) Hence, J n ( x, t ) := Z π − π e − q n ( x,t,θ ) d θ 2 π . T o establish Prop osition 3.1, w e need to sho w the conv ergence to s 2 2 of the log-moment generating function Ψ n ( s ) := ln E  exp  s  K G,n − a n b n  . Again, the notation closely follo ws [6] and Ψ n ( s ) is unrelated to the function Ψ( s ) temp orarily in tro duced for the purp oses of pro ving Prop osition 2.2. W e define the t w o sequences t n := W − 1 G, 0  n x  , and x n := xe s b n . By Lemma 2.3, we immediately deduce the asypm totics t n ∼ ( x K G ) 1 α G × α −  m G − 1 α G  G × n − 1 α G × (ln n ) m G − 1 α G , (9) and in particular t n → 0 + , when n → ∞ . W e also ha ve Ψ n ( s ) = − sa n b n + ln P n ( x n , t n ) − ln P n ( x, t n ) + R n ( s ) with R n ( s ) := ln  J n ( x n , t n ) J n ( x, t n )  . (10) Note that the constant sequence equal to x can also b e viewed as the x n sequence, in the sp ecial case when s = 0. The saddle p oint metho d ` a la Hayman requires c hoosing the contour or the radius r = e − t , in order to minimize the maximum o ver the contour of the mo dulus of the integrand. This gives the equation d d t ( nt + xW G, − 1 ( t )) = 0 , whic h is precisely solved b y our choice for t n . Th us the optimization is exact for the integral giving H G,n ( x ), but only approximate for the in tegral giving H G,n ( x n ). An easy calculation shows, as in [6, § 5], Ψ n ( s ) = A n s + B n s 2 2 + x  e s b n − 1 − s b n − s 2 2 b 2 n  W G, − 1 ( t n ) + R n ( s ) , (11) 15 with A n := xW G, − 1 ( t n )( t n ) − a n b n , (12) B n := xW G, − 1 ( t n ) b 2 n . (13) W e will split the integral J n ( x n , t n ) in to t w o pieces: a ma jor arc region of in tegration R ma j := { θ ∈ ( − π , π ) | | θ | ≤ t n } , and a minor arc region of in tegration R min := { θ ∈ ( − π , π ) | | θ | > t n } . W e will use separate estimates for these t wo regions, in order to determine asymptotics for the in tegral J n ( x n , t n ). 4. Major and minor arc estima tes F rom the definition (8), we deduce Re q n ( x n , t n , θ ) = x n ∞ X δ =1 e − δ t n δ a δ ( G ) (1 − e iδ θ ) , (14) = 2 x n ∞ X δ =1 a δ ( G ) δ e − δ t n sin 2  δ θ 2  . W e let N n :=  π t n  , and see that for θ ∈ R ma j w e ha ve Re q n ( x n , t n , θ ) ≥ 2 x n N n X δ =1 a δ ( G ) δ e − δ t n sin 2  δ θ 2  , ≥ 2 x n θ 2 π 2 e π N n X δ =1 δ a δ ( G ) , where w e use the conv exit y estimate sin u ≥ 2 u π for 0 ≤ u ≤ π 2 , with u =   δ θ 2   . This is b ecause 1 ≤ δ ≤ N n ≤ π t n , and | θ | ≤ t n b y h yp othesis. W e also used δ t n ≤ π to b ound e − δ t n from b elo w by e − π . Since N n ∼ π t n , w e deduce from Prop osition 2.2 with β = 1, 2 x n θ 2 π 2 e π N n X δ =1 δ a δ ( G ) ∼ 2 xθ 2 π α G − 1 γ G e π ( m G − 1)!( α G + 1) × t − α G − 1 n × ( − ln t n ) m G − 1 , ≥ η ma j θ 2 t − α G − 1 n ( − ln t n ) m G − 1 , for some constant η ma j > 0 (p ossibly dep enden t on x ) for all n large enough. W e now define λ n := ( x n W G, 1 ( t n )) − 1 2 , 16 whic h satisfies λ n ∼ ( xα G K G ) − 1 2 × t α G +1 2 n × ( − ln t n ) − ( m G − 1 2 ) , (15) b y (6). W e will do the c hange of v ariable θ = λ n Θ in the in tegral ov er the ma jor arc region, i.e., write Z R ma j e − q n ( x,t,θ ) d θ 2 π = λ n 2 π Z R f n (Θ) dΘ , where f n : R → C is defined by f n (Θ) := 1 l {| Θ | ≤ t n λ − 1 n } e − q ℓ,n ( x n ,t n ,λ n Θ) , with n large enough so that t n < π . Note that since α G > 1, b y (15), w e hav e lim n →∞ t n λ − 1 n = ∞ , so the effective domain of in tegration go es to the entire real line. F rom form ula (8), and b ecause of the definition of x n , t n and λ n , we hav e after the change of v ariables, q n ( x n , t n , λ n Θ) = i Θ λ n n (1 − e s b n ) + Θ 2 2 + Err , where the error term is giv en b y Err = − x n X δ ≥ 1 e − t n δ δ a δ ( G ) T ( iδ λ n Θ) , with the notation T ( v ) := e v − 1 − v − v 2 2 . By T a ylor’s formula with integral remainder one easily sees that w e hav e the b ound T ( v ) ≤ | v | 3 6 , if v is pure imaginary . By applying this to v = iδ λ n Θ in the formula for the error term, w e obtain | Err | ≤ 1 6 x n Θ 3 λ 3 n W G, 2 ( t n ) . By (7) and (15), and since α G > 1, w e see that lim n →∞ λ 3 n W G, 2 ( t n ) = 0 . Hence p oin twise in Θ ∈ R , the error term go es to zero. F rom (9) we see that ( − ln t n ) ∼ ln n α G . So b y com bining (9) and (15), w e obtain the more explicit asymptotics in term of n giv en b y λ n ∼ 1 √ α G × α G −  m G − 1 2 α G  × ( x K G ) 1 2 α G × n −  α G +1 2 α G  × (ln n ) m G − 1 2 α G . (16) T ogether with the h yp othesis on b n in Prop osition 3.1, this gives lim n →∞ nλ n b n = √ α G − 1 . As a result, we hav e p oint wise in Θ ∈ R , lim n →∞ f n (Θ) = exp  is Θ √ α G − 1 − Θ 2 2  . Recall that for n large enough, and for | Θ | ≤ t n λ − 1 n , w e ha v e the ma jor arc estimate q n ( x n , t n , λ n Θ) ≥ η ma j Θ 2 λ 2 n t − α G − 1 n ( − ln t n ) m G − 1 . 17 Pic k some constan t κ > 0 such that κ < η ma j × ( xα G K G ) − 1 , then from (15), we see that q n ( x n , t n , λ n Θ) ≥ κ Θ 2 . W e sho wed that for n large enough, we ha v e for all Θ ∈ R , | f n (Θ) | ≤ e − κ Θ 2 . W e can no w apply the dominated conv ergence theorem: lim n →∞ Z R f n (Θ) dΘ = Z R e is Θ √ α G − 1 − Θ 2 2 dΘ = √ 2 π e − ( α G − 1) s 2 2 , b y the form ula for the F ourier transform of a Gaussian. W e no w consider angles θ in the minor arc region R min . W e go back to (14) and use the at least linear gro wth h yp othesis a δ ( G ) ≥ cδ for some constant c > 0. This allo ws us to follo w [6, § 4.2] almost verbatim. W e ha v e Re q n ( x n , t n , θ ) ≥ c x n Re ρ ( t n , θ ) , with ρ ( u, θ ) := ∞ X k =1 e − ku  1 − e ikθ  , = e u ( e u − e − iθ − e u e iθ + 1) ( e u − 1)( e 2 u − 2 e u cos θ + 1) . As explained in [6, § 4.2], w e ha ve when θ ∈ R min , the low er b ound and asymptotic Re q n ( x n , t n , θ ) ≥ x n × e t n ( e t n + 1) e t n − 1 × 1 − cos( t n ) e 2 t n − 2 e t n cos( t n ) + 1 ∼ x 2 t n . In view of (9), there exists a constant η min > 0 (p ossibly dep ending on x ) suc h that for n large enough and all θ ∈ R min , Re q n ( x n , t n , θ ) ≥ η min n 1 α G (ln n ) −  m G − 1 α G  . This implies the decay estimate     Z R min e − q ℓ,n ( x n ,t n ,θ ) d θ 2 π     ≤ exp  − η min n 1 α G (ln n ) −  m G − 1 α G   . The latter is negligible with resp ect to λ n whic h follo ws a p ow er la w times a logarithm as sho wn in (16). As a result, the asmyptotics of the full in tegral is determined b y the p ortion due to the ma jor arc region R ma j : J n ( x, t ) ∼ Z R ma j e − q ℓ,n ( x n ,t n ,θ ) d θ 2 π , ∼ λ n 2 π √ 2 π e − ( α G − 1) s 2 2 , ∼ e − ( α G − 1) s 2 2 √ 2 π α G × α G −  m G − 1 2 α G  × ( x K G ) 1 2 α G × n −  α G +1 2 α G  × (ln n ) m G − 1 2 α G . (17) 18 5. Completion of the proof of the main theorem W e now finish the pro of of Prop osition 3.1 and then that of Theorem 1.2. By applying (17) twice, for the given s ∈ R , and then for the particular case s = 0 (where x n b ecomes the constan t sequence equal to x ), w e see that the remainder (10) satisfies lim n →∞ R n ( s ) = − ( α G − 1) s 2 2 . F rom the hypotheses on the sequences a n and b n , we immediately see that A n , defined in (12), is such that lim n →∞ A n = 0 . F rom (4), (9), and the previous remark that ( − ln t n ) ∼ ln n α G , w e get xW G, − 1 ( t n ) ∼ 1 ( α G − 1) × α G −  m G − 1 α G  × ( x K G ) 1 α G × n α G − 1 α G × (ln n ) m G − 1 α G . The h yp othesis on b n then implies that B n , defined in (13), satisfies lim n →∞ B n = α G . Since it asymptotically contains, in comparison to B n , an extra factor 1 b n whic h go es to zero, the expression x  e s b n − 1 − s b n − s 2 2 b 2 n  W G, − 1 ( t n ) also go es to zero as n → ∞ . As a result, the log-momen t generating function giv en in (11) satisfies lim n →∞ Ψ n ( s ) = s 2 2 , and Prop osition 3.1 is established. □ The pro of of the main theorem now follows the same steps as in [6]. W e make tw o rounds of application of Prop osition 3.1. W e first pic k a n := xW G, − 1  W − 1 G, 0  n x  , b n := 1 p α G ( α G − 1) × α G −  m G − 1 2 α G  × ( x K G ) 1 2 α G × n α G − 1 2 α G × (ln n ) m G − 1 2 α G . Then Prop osition 3.1 gives the con v ergence of the momen t generating function of the random v ariables K G,n − a n b n to that of the standard Gaussian. By the contin uit y theorem of Curtiss (see [6, Thm 1.2]), this implies the con vergence of momen ts. F rom the conv ergence of the first t w o momen ts, we immediately conclude that the sequences given by mean and the standard deviation satisfy the h yp otheses of Prop osition 3.1. W e then redefine our a n and b n sequences as a n := E K G,n , b n := q V ar( K G,n ) . W e then apply Prop osition 3.1 again, follow ed b y the Curtiss contin uit y theorem, and finally conclude the pro of of Theorem 1.2. □ 19 6. Outlook It would b e in teresting to examine the statistics of K G,n , for n large, when the function a n ( G ) gro ws faster than a p olynomial. Do es the v ariance stay b ounded? W ould a limit theorem to w ards a discrete distribution lik e P oisson b e more appropriate than a CL T? In [6], w e mentioned right-angled Artin groups G as a direction of generalization aw a y from the Ab elian case. Ho wev er, as so on as one remov es one edge in the relev ant graph, the a n ( G ) gro ws to o fast [7]. The author was th us led to the study for the even “more commutativ e” case of nilp oten t groups in the present article. W e used the ad ho c tec hniques in [6], but another p ossibilit y w ould to see if the CL T from [33] is applicable. This would require c hec king the Ha yman log-admissibility condition whic h is a nontrivial task we lea ve to future w ork. It would b e in teresting to in v estigate, if only numerically for now, log-conca vit y of the n um b ers A ( G, n, k ) with resp ect to k , i.e., the inequalites A ( G, n, k ) 2 ≥ A ( G, n, k − 1) A ( G, n, k + 1) , (18) for 2 ≤ k ≤ n − 1, and for a v ariet y of groups G , e.g., T -groups. F rom (3), one easily sees that A ( G, n, k ) = n ! k ! X n 1 ,...,n k ≥ 1 1 l { n 1 + · · · + n k = n } × a n 1 ( G ) · · · a n k ( G ) n 1 · · · n k . This is the partition function of a p olymer gas, in the language of statistical mechanics [21]. Moreo v er, the relev ant p olymer activities only dep end on the size of the p olymers, and not on their shap e or geometrical features. This transformation relating the a n ( G ) to the A ( G, n, k ) is also called a Bell transform [35]. The author lik es to see this as a kind of non-linear av eraging of the a n ( G ). F or G = Z ℓ , the log-concavit y w as conjectured by Heim and Neuhauser for ℓ = 2 [24]. This log-concavit y conjecture was extended to all ℓ ≥ 2 b y the author [2]. In the remark able work [39] concerning the ℓ = 2 case, Starr w as able to establish (18) in the n → ∞ limit with k fixed, when k ≥ 3. How ev er, for ℓ = 2 and k = 2, Starr show ed that (18) fails for n large. Th us the conjectures b y Heim-Neuhauser and the author m ust b e amended to exclude v alues of k that are to o low. It seems the men tioned a v eraging must b e non-linear enough in order to see log-concavit y . Finally , if log-conca vity is supp orted by n umerical evidence, it w ould b e interesting to lo ok at what n umber-theoretical consequences this log-concavit y could hav e. Ac kno wledgemen ts: F or useful correspondence or quic k in p erson discussion, w e thank Mikhail Ersho v, Ofir Goro detsky , Bernhard Heim, Thomas Kob erda, Vyac hesla v Krushk al, Michael Magee, Markus Neuhauser, Doron Puder, and Gerald T enen baum. W e thank Ramon v an Handel for insigh tful discussions on subgroup growth and random cov erings. W e thank Paul Anderson for the related ongoing work [4]. Last but not least, w e thank Shannon Starr for the collab oration [6] where man y of the techniques used in this article were developed. References [1] A. Ab desselam, F eynman diagrams in algebraic combinatorics. S ´ em. Lothar. Com bin. 49 (2002/04), Art. B49c, 45 pp. (electronic). [2] A. Ab desselam, Log-concavit y with resp ect to the n umber of orbits for infinite tuples of commuting p erm utations. Ann. Comb. 29 (2025), no. 2, 563–573. 20 [3] A. Ab desselam, Pro of of a conjecture by Starr and log-conca vit y for random commuting p erm utations. 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Box 400137, University of Virginia, Charlottesville, V A 22904-4137, USA Email addr ess : malek@virginia.edu 22