A new source of purely finite matricial fields

A NEW SOUR CE OF PUREL Y FINITE MA TRICIAL FIELDS DA VID GAO, SRIV A TSA V KUNNA W ALKAM ELA Y A V ALLI, AAREY AN MANZOOR, AND GREGOR Y P A TCHELL Abstract. A coun table group G is said to be matricial field (MF) if it admits a strongly conv erging sequence of approximate homomorphisms into matrices; i.e, the norms of p olynomials con v erge to those in the left regular represen- tation. G is then said to b e purely MF (PMF) if this sequence of maps into matrices can be c hosen as actual homomorphisms. G is further said to b e pur ely finite field (PFF) if the image of eac h homomorphism is finite. By developing a new op erator algebraic approach to these problems, we are able to prov e the following result bringing several new examples into the fold. Supp ose G is a MF (resp., PMF, PFF) group and H < G is separable (i.e., H = ∩ i ∈ N H i where H i < G are finite index subgroups) and K is a residually finite MF (resp., PMF, PFF) group. If either G or K is exact, then the amalgamated free pro duct G ∗ H ( H × K ) is MF (resp., PMF, PFF). Our work has sev eral applications. Firstly , as a consequence of MF, the Brown–Douglas–Fillmore semigroups of many new reduced C ∗ -algebras are not groups. Secondly , we obtain that arbitrary graph pro ducts of residually finite exact MF (resp., PMF, PFF) groups are MF (resp., PMF, PFF), yield- ing a significant generalization of the breakthrough work of Magee–Thomas. Thirdly , our work resolves the op en problem of proving PFF for 3-manifold groups. More generally all groups that virtually embed into RAAGs are PFF. Prior to our w ork, PFF was not kno wn even in the case of free products. Our results are of geometric significance since PFF is the property that is used in Antoine Song’s approac h in the theory of minimal surfaces. for Vidhya R anganathan 1. Introduction 1.1. Matricial field. Understanding finite dimensional approximations for op er- ator algebras allows one to effectively imp ort in tuition from the b ehavior in finite dimensions to understand infinite dimensional op erator algebras and vice versa. An excellent example of the forwards direction is the recent revolutionary w ork on the Connes embedding problem [JNV + 21]. On the other hand, V oiculescu’s in tro duction of free probability theory motiv ated b y his study of the free group factors, in particular his cornerstone asymptotic freeness result [V oi91, VDN92], offers great testimon y to the p o w er of the bac kwards direction. The av ailability of matrix models that con v erge w eakly to tracial noncommutativ e distributions in v on Neumann algebras has also yielded striking insigh ts into their structure, via free en tropy theory [V oi02, Hay22]. The subsequent groundbreaking work of Haagerup and Thorb jørnsen [HT05] op ened up the p ossibility of extending the “weak con ver- gence” of matrix mo dels into the setting of C ∗ -algebras where one would demand 1 2 D. GAO, S. KUNNA W ALKAM ELA Y A V ALLI, A. MANZOOR, AND G. P A TCHELL “strong con vergence” in op erator norms. The endeav or of establishing strong con- v ergence has recently gained prominence in b oth random matrix theory and op era- tor algebras, and has pav ed the wa y for solutions to several long standing problems in multiple areas of mathematics and applied mathematics [vH25, Mag25]. Our present article furthers the study of strong con v ergence. The matricial field (MF) property of countable groups is due to Blac k adar and Kirc hberg [BK97]. This prop ert y asks for finite dimensional approximations; i.e., matrix models for reduced group C ∗ -algebras in the strong sense. T o b e precise, we sa y G is MF if for all finite sets F ⊆ G and  > 0, there are d ∈ N and a function u : G → U ( M d ( C )) with (1) ∥ u g h − u g u h ∥ <  for all g , h ∈ F , (2) | τ ( u g ) | <  for all g ∈ F \ { 1 } , and (3)     P g ∈ F c g u g   −   P g ∈ F c g λ g   C ∗ r ( G )   <  for all ( c g ) g ∈ F ⊆ C with max g ∈ F | c g | ≤ 1, where λ g refers to the unitary asso ciated to g ∈ G in the left regular represen tation. In other words, G is MF if there exists an ultrafilter U on N and a trace-preserving ∗ -homomorphism from the reduced C ∗ -algebra, π : C ∗ r ( G ) → Q U M n ( C ), where the target C ∗ -algebra is the ultrapro duct of matrix algebras with resp ect to the op erator norm. F urthermore, G is said to b e pur ely MF (PMF) if the maps u from ab o v e can b e chosen as actual homomorphisms. G is said to b e a pur ely finite field (PFF) if the image of the homomorphisms are finite at each stage. G is further said to be a pur ely p ermutation field (PPF) if in fact these homomorphisms are homomorphisms to finite p ermutation groups comp osed with standard irreducible represen tations of p ermutation groups. Note that the notations PMF and PPF are b orro w ed from [Mag25]. The relationship b etw een these notions is transparen t and is given b elow: PPF = ⇒ PFF = ⇒ PMF = ⇒ MF . Outside of intrinsic interest in random matrix theory , these prop erties hav e phe- nomenal applications. MF has applications to op erator algebras. If a C ∗ -algebra is MF but not quasidiagonal, then A has an extension by the compact operators K whic h is not inv ertible in the sense of Brown, Douglas and Fillmore [BDF77] (see for instance the remark at the end of Section 2 in [See12]). In particular if a group is non amenable (hence not quasidiagonal [Had87]) and MF, then E xt ( C ∗ r ( G )) is not a group. PMF has certain geometric applications [MT23], including sp ectral gaps of certain Laplacians acting on vector bundles. PFF has remark ably found use in the theory of m inimal surfaces through w ork of A. Song in his approach [Son25]. PFF is also interestingly in the spirit of soficity [Pes08] for algebras: it is not just a finite-dimensional approximation of the reduced group C ∗ -algebra, but even a finite appro ximation. PPF is the most desirable prop erty here not only b ecause it encompasses all of the ab o v e, but also b ecause it pla ys a fundamen tal role in deep applications to the study of optimal sp ectral gaps of graphs and hyperb olic manifolds [BC19, HM23]. PPF is how ever quite difficult to access, esp ecially b e- cause it fails in certain simple examples (see the discussion b efore Remark 1.5). The main nov elty of the present article is that despite this it is still p ossible, in significan t generality , to bypass PPF and access PFF instead. F or further details and references concerning these notions we p oint the reader to the b eautiful surv eys [Mag25, vH25]. A NEW SOURCE OF PUREL Y FINITE MA TRICIAL FIELDS 3 Imp ortan tly , these prop erties are muc h harder to prov e than weak er representabil- it y prop erties such as soficity or Connes em b eddabilit y [Pes08], b ecause it inv olv es establishing strong con vergence. Despite the v ast collection of examples of soficity and h yp erlinearit y , at the moment only a handful of examples of groups are known to b e MF/PMF/PFF/PPF. Moreov er, it is noteworth y to p oint out that pro ving suc h prop erties in eac h of these cases has been profoundly difficult and needed deep insigh ts. The current list of examples includes MF for amenable groups [TWW17]; PPF for free groups [BC19]; PPF for limit groups [LM25]; PMF for right angled Artin groups (RAAGs) [MT23]; MF for crossed pro ducts b y free groups acting on amenable groups [RS19]; MF for G 1 ∗ H G 2 where G 1 , G 2 are amenable and H is a normal subgroup [Sch24]. These prop erties are easily seen to b e closed under tak- ing subgroups and MF/PMF/PFF are closed under taking finite index ov ergroups (Lemma 7.1 in [LM25], see also Lemma 2.4). Being c losed under finite index o v er- groups seems op en in the case of PPF. It is curren tly kno wn that MF/PMF is also closed under free pro ducts [CM14, Hay15]. Moreo v er, MF/PMF/PFF is also closed under direct pro ducts provided one of the groups is exact [HS10]. Strikingly , PPF is necessarily not closed under direct pro ducts, see [Mag25] for a pro of that F 2 × F 2 × F 2 is not PPF. Despite all of these results, the situation for wider natural families of groups, in particular for amalgamated free pro ducts, has remained a c hallenging op en problem. 1.2. Main results. In this pap er w e are able to make significan t new progress and expand on the collection MF and notably also PMF and PFF groups by developing a new C ∗ -algebraic approach to the problem. Our main result is the following: Theorem 1.1. Supp ose G is an MF (r esp., PMF, PFF) gr oup and H < G is sep ar able (i.e., H = ∩ i ∈ N H i wher e H i < G ar e finite index sub gr oups). L et L b e a r esidual ly finite MF (r esp., PMF, PFF) gr oup such that either G or L is exact. Then the amalgamate d fr e e pr o duct G ∗ H ( H × L ) is MF (r esp., PMF, PFF). Our approac h is inspired b y recent progress on selflessness which is a form of in trin- sic C ∗ -free indep endence in ultrap ow ers [Rob25, AGKEP25, Oza25], in particular the work of Oza wa (see Section 4.6 of [BO08]). In order to demonstrate the p ow er of our result, we assemble a few corollaries. Firstly , we observe that there is an isomorphism G ∗ H ( H × Z ) ∼ = ∗ H G ⋊ Z where ∗ H G denotes the infinite group double of G ov er H , indexed by elements of Z , and Z acts on it by p erm uting the copies of G in accordance with the left multiplication action of Z on itself. W e also recall that, if G and H are PFF and one of them is exact, then G × H is PFF (Lemma 2.4). PFF groups are also automatically residually finite. This yields: Corollary 1.2. L et G b e a MF (r esp., PMF, PFF) gr oup and H < G b e a sep ar able sub gr oup. Then the gr oup double G ∗ H G is MF (r esp., PMF, PFF). Mor e over, if G and H ar e PFF gr oups such that one of them is exact, then G ∗ H is PFF. W e p oin t out that strikingly even the case of free pro ducts remained op en for the PFF property un til our w ork. Outside of families of LERF groups (see a list of such examples in [Gao25]), a nice instance of such inclusions H < G which can b e fed in to our Corollary 1.2 is quasi-conv ex subgroups of cubulated hyperb olic groups. Indeed, cubulated hyperb olic groups are subgroups of RAAGs [Ago13] and hence PFF (our Corollary 1.3 b elo w), and quasi-conv ex subgroups are separable [HW08]. 4 D. GAO, S. KUNNA W ALKAM ELA Y A V ALLI, A. MANZOOR, AND G. P A TCHELL W e also p oint out that even soficity or Connes embeddability is unknown b eyond the generality we consider in such amalgamated free pro ducts. Another natural setting for separability is in the world of graph pro ducts [HW99]. Corollary 1.3. Arbitr ary gr aph pr o ducts of exact r esidual ly finite MF (r esp., PMF, PFF) gr oups ar e MF (r esp., PMF, PFF). Our result ab o v e offers a significant generalization of the breakthrough work of Magee–Thomas [MT23]. W e emphasize that even in the case of RAAGs, our pro of is different. While w e still use the MF/PMF/PFF property of F n [HT05, BC19] as a blac k b o x in our pro of, our approac h a voids the use of other random matrix theory results [CGP22], as well as the Crisp–Laca universalit y theorem [CL02] which are b oth crucial in [MT23]. More imp ortan tly , the reac h of our results is higher and can handle the PFF prop ert y for RAAGs, whic h has b een an op en problem in the field 1 . This particular result is of significance b ecause this is exactly the ingredient that go es into the breakthrough work of A. Song on minimal surfaces in Euclidean unit spheres [Son25]. A. Song’s pap er accesses PFF via the PPF result of Bordenav e– Collins [BC19], and the follow up paper [ALST25] accesses PFF for surface groups via Louder–Magee’s PPF for surface groups [LM25]. Ho w ev er, our results op en the do or to v ast new families of groups, without having to go through PPF. This in particular includes 3-manifold groups and a plethora of other examples that virtually embed into RAAGs [Ago13, HW08]. W e do cumen t this below. Corollary 1.4. F undamental gr oups of close d 3-manifolds ar e PFF. Then G is PFF. Mor e gener al ly, al l gr oups that virtual ly emb e d into RAAGs ar e PFF. W e now discuss optimalit y asp ects of our results. First note that the exactness is a rather essential ingredient, in for instance Corollary 1.3. Indeed even in the tensor product case it is not known if MF is preserv ed in full generality [HS10]. A natural question arises of whether our results can extend to accessing PPF. W e p oin t out that this is in fact not p ossible. F or instance, our Corollary 1.3 simply cannot generalize to PPF as stated b ecause as mentioned earlier, Magee has shown that the group F 2 × F 2 × F 2 is not PPF [Mag25], and this is indeed a RAAG. F or this reason PFF seems to b e the b est one can hop e for in these general families of groups. The challenge with PPF is that it is not amenable to b eing stable under direct products (or finite index ov ergroups), due to the follo wing fundamen tal issue: tensoring do es not preserve the image b eing in the orthogonal complemen t of the in v ariant vectors for p ermutation matrices. Remark 1.5. W e conclude with a minor remark that our w ork also provides an alternativ e pro of of PFF for limit groups also (see Remark 2.6). It is imp ortant to clarify how ev er that the work [LM25] pro ves PPF for limit groups and this cannot b e addressed curren tly by our metho d. The proof of this in our work combines our Theorem 1.1 and the fact that limit groups arise as iterated extensions of cen tralizers (Theorem 4.2 of [CG05]), and Hall’s theorem for limit groups [Wil08]. W e also p oin t out that the pro of of Corollary 1.3 follows from the fact that every graph pro duct is constructed in an iterated fashion using the amalgamated free pro ducts in Theorem 1.1, and additionally the amalgams are separable b ecause they are retracts (Lemma 3.9 of [HW99]). W e also remark that our approach migh t 1 W e thank Ramon v an Handel for sharing with us the statemen t of this op en problem. A NEW SOURCE OF PUREL Y FINITE MA TRICIAL FIELDS 5 also naturally b e applied to get more examples of MF/PMF/PFF groups among the family of graph wreath pro ducts [GKEP24]. 1.3. Ac kno wledgmen ts. W e thank M. Junge for an insigh tful question that stim- ulated this w ork during the workshop “Graduate mini-school on sofic groups and Connes em b edding problem” at UT Austin during March 18–22, 2026. W e thank UT Austin and the organizers for providing a stimulating environmen t to work in. W e are indebted to R. v an Handel for sev eral long corresp ondences and his extremely generous feedback and supp ort. W e thank D. Jekel and N. Oza w a for helpful comments and corrections. W e also thank B. Hay es and F. F ournier-F acio for helpful conv ersations regarding references and M. Linton for his suggestion of a quic k pro of of Lemma 2.5. The second author is grateful to U. Bader for his en- couragemen t and helpful conv ersations. This w ork was supp orted partially by the second author’s NSF grant DMS 2350049. The fourth author was supported b y the Engineering and Physical Sciences Research Council (UK), gran t EP/X026647/1. Op en Access and Data Statemen t. F or the purp ose of Op en Access, the au- thors hav e applied a CC BY public copyrigh t license to any Author Accepted Man- uscript (AAM) v ersion arising from this submission. Data sharing is not applicable to this article as no new data w ere created or analyzed in this work. 2. Proof of main resul t W e will use standard notations in C ∗ -algebras, in particular Chapter 4 of the bo ok [BO08], which is a great reference for a reader who is familiar with basic asp ects of C ∗ -algebra theory . Unless otherwise mentioned, amalgamated free pro ducts of C ∗ -algebras equipped with conditional exp ectations will b e in the reduced sense [V oi85] 2 . Additionally , all ultrapro ducts will b e with resp ect to the op erator norm, and all tensor pro ducts will b e minimal. W e recall the following for notational purp oses. Definition 2.1. Let B ⊂ A b e an inclusion of C ∗ -algebras, and E : A → B be a faithful conditional exp ectation. Then define the T o eplitz algebr a of E to be the univ ersal C ∗ -algebra T ( E ) := C ∗ ⟨ a ∈ A , T   T ∗ T = 1 , T ∗ aT = E ( a ) ⟩ . T is called the cr e ation op er ator . The following Theorem 2.2 is due to Shlyakh tenk o in [Shl99] in the von Neumann algebra setting (see also [FMR03]). The pro of in the C ∗ -setting is more or less iden tical. Nevertheless, w e provide a pro of for the conv enience of the reader (fol- lo wing [BO08] wherein this result also app ears). Our inspiration for employing this result in our context comes from the recent outburst of developmen ts surrounding selfless C ∗ -algebras [Rob25, AGKEP25], esp ecially the pro of technique of Ozaw a in [Oza25] and also the up coming key work [JKEPR]. The second and fourth authors are grateful to M. Junge and L. Rob ert for the collab oration on [JKEPR]. 2 W e alert the reader to not b e confused with MF for full C ∗ -algebras or for full amalgamated free pro ducts (see for instance [Shu26]) as the problems and tec hniques in that situation are of a very different nature. 6 D. GAO, S. KUNNA W ALKAM ELA Y A V ALLI, A. MANZOOR, AND G. P A TCHELL Before stating and pro ving Theorem 2.2, we emphasize that the b eautiful gauge- in v ariant uniqueness theorem (this is explicitly [BO08, 4.6.18]), is used in the pro of of Theorem 2.2. W e highly encourage the reader to look at Section 4.6 in the treatise [BO08] which offers a clean treatment of these asp ects. Theorem 2.2. L et the notation b e as ab ove. Then the C ∗ -sub algebr a gener at e d by A and 1 2 ( T + T ∗ ) in T ( E ) is isomorphic to A ∗ B ( B ⊗ C [ − 1 , 1]) , wher e 1 2 ( T + T ∗ ) is the semicir cular in C [ − 1 , 1] . Pr o of. First consider A ∗ B ( B ⊗ T ) where T is the T o eplitz algebra. Let S b e the generating isometry in T . Then for a ∈ A , S ∗ aS = E ( a ). Hence there is a map π : T ( E ) → A ∗ B ( B ⊗ T ) which restricts to identit y on A and sends T 7→ S . This map is surjective as A ∗ B ( B ⊗ T ) is generated by A and T . W e will prov e π is injective using the gauge-in v ariant uniqueness theorem for T o eplitz-Pimsner algebras [BO08, 4.6.18]. F or this, note that the canonical con- ditional exp ectation E ′ : A ∗ B ( B ⊗ T ) → A acts by id ⊗ ω on the second factor, where ω is the v acuum state on the T o eplitz algebra. So w e ha v e E ′ ( S aS ∗ ) = E ′ ( S E ( a ) S ∗ ) + E ′ ( S ( a − E ( a )) S ∗ ) = E ( a ) E ′ ( S S ∗ ) = 0 , where we used that S commutes with E ( a ) ∈ B . So π ( A ) ∩ span { a 1 S b 1 b ∗ 2 S ∗ a ∗ 2 : a 1 , b 1 , b 2 , a 2 ∈ A } = { 0 } , and so π satisfies the h ypothesis of the gauge-inv ariant uniqueness theorem. Restricting to the subalgebra generated b y A and 1 2 ( T + T ∗ ) in T ( E ) giv es the result. □ The following is the key argument in our pro of. Lemma 2.3. L et G b e a gr oup and H a sub gr oup with H = T i ∈ N H i for some de cr e asing se quenc e of sub gr oups H i . L et K i = T g ∈ G g H i g − 1 b e the normal c or e of H i . L et U b e a fr e e ultr afilter on N . Then ther e is a tr ac e-pr eserving emb e dding of C ∗ -algebr as: C ∗ r ( G ∗ H ( H × Z ))  → Y U C ∗ r ([ G/K i ∗ H i /K i ( H i /K i × Z )] × G ) . Mor e over, this emb e dding lifts to a se quenc e of gr oup homomorphisms G ∗ H ( H × Z ) → [ G/K i ∗ H i /K i ( H i /K i × Z )] × G. Pr o of. Let E : C ∗ r ( G ) → C ∗ r ( H ) and E i : C ∗ r ( G/K i ) → C ∗ r ( H i /K i ) b e the resp ec- tiv e conditional exp ectations. Let q i : C [ G ] → C ∗ r ( G/K i ) b e the map induced from the quotient map G → G/K i . De note by T the creation op erator in T ( E ) and by T i the creation op erator in T ( E i ). T ake C [ G ] → Q U T ( E i ) ⊗ C ∗ r ( G ) to b e the map that sends λ g to ( q i ( λ g ) ⊗ λ g ) U for g ∈ G , where we recall T ( E i ) contains a copy of C ∗ r ( G/K i ). By F ell’s absorption principle, this is isometric if C [ G ] is giv en the reduced norm, and so extends to an em b edding C ∗ r ( G )  → Q U T ( E i ) ⊗ C ∗ r ( G ). Identifying the image of this embedding with C ∗ r ( G ), it is easy to c hec k that the conditional exp ectation E under this iden tification sends ( q i ( λ g ) ⊗ λ g ) U to ( q i ( E ( λ g )) ⊗ λ g ) U for g ∈ G . Now note ( T ∗ i q i ( λ g ) T i ⊗ λ g ) U = ( E i ( q i ( λ g )) ⊗ λ g ) U = ( q i ( E ( λ g )) ⊗ λ g ) U . A NEW SOURCE OF PUREL Y FINITE MA TRICIAL FIELDS 7 The latter equality can b e verified easily using the fact that H i decreases to H . Hence, the isometry ( T i ⊗ 1) U satisfies the univ ersal property of the creation op erator in T ( E ) and this gives a *-homomorphism T ( E ) → Y U T ( E i ) ⊗ C ∗ r ( G ) . This map is in fact an embedding. This follo ws from the gauge-in v ariant uniqueness theorem for T o eplitz–Pimsner algebras [BO08, Theorem 4.6.18]. Indeed, note that ( E ′ i ⊗ id C ∗ r ( G ) ) U defines a conditional exp ectation from Q U T ( E i ) ⊗ C ∗ r ( G ) onto a subalgebra con taining the image of C ∗ r ( G ), where E ′ i : T ( E i ) → C ∗ r ( G/K i ) are the canonical conditional exp ectations. The condition needed for the gauge-inv ariant uniqueness theorem easily follows, as in the pro of of Theorem 2.2. No w, by Theorem 2.2, restricting to C ∗ ( G, 1 2 ( T + T ∗ )) ⊂ T ( E ) yields an embedding C ∗ r ( G ) ∗ C ∗ r ( H ) ( C ∗ r ( H ) ⊗ C [ − 1 , 1])  → Y U [ C ∗ r ( G/K i ) ∗ C ∗ r ( H i /K i ) ( C ∗ r ( H i /K i ) ⊗ C [ − 1 , 1])] ⊗ C ∗ r ( G ) . That this is trace-preserving can b e easily v erified. Since 1 2 ( T + T ∗ ) is sen t to ( 1 2 ( T i + T ∗ i )) U , the map acts as the iden tit y map on C [ − 1 , 1], so the result no w follo ws b y noting C ∗ r ( Z ) embeds into C [ − 1 , 1] trace-preservingly . F or the second part, w e note that the map is indeed induced by the sequence of homomorphisms G ∗ H ( H × Z ) → [ G/K i ∗ H i /K i ( H i /K i × Z )] × G whic h send g ∈ G to ( g K i , g ) and n ∈ Z to ( n, e ). □ W e note the following fact whic h is certainly well kno wn to experts, see for instance Lemma 7.1 of [LM25]. F or the b enefit of the reader we include a pro of. Lemma 2.4. (1) Virtual ly fr e e gr oups ar e PFF; (2) If G and H ar e PMF and G is exact, then G × H is PMF. If they ar e in addition PFF, then the pr o duct is PFF as wel l. Pr o of. F or (1), free groups are PPF by [BC19] and so PFF. The result follows b y considering induced representations. T o be precise, assume H is PFF and G con tains H as a finite-index subgroup. Then we ma y take a sequence of strongly con verging representations σ n : H → M d ( n ) ( C ) that quotien ts through finite groups H n . Then the induced representations φ n : G → M [ G : H ] ( M d ( n ) ( C )) conv erge strongly . F urthermore, the range of φ n is con tained within the following set of [ G : H ]-b y-[ G : H ] matrices with entries in M d ( n ) ( C ): { A ∈ M [ G : H ] ( M d ( n ) ( C )) : there is a p ermutation matrix P ∈ M [ G : H ] ( C ) s.t. A ij = 0 whenever P ij = 0 and A ij ∈ range( σ n ) whenever P ij = 1 } . It is easy to see that the set ab ov e is finite, so the result follo ws. 8 D. GAO, S. KUNNA W ALKAM ELA Y A V ALLI, A. MANZOOR, AND G. P A TCHELL Item (2) for PMF follo ws b y noting that, as G is exact, the following is short exact sequence, 0 → M n M d ( n ) ( C ∗ r ( G )) → Y n M d ( n ) ( C ) ⊗ C ∗ r ( G ) → Q n M d ( n ) ( C ) L n M d ( n ) ( C ) ⊗ C ∗ r ( G ) → 0 . The middle term embeds in to Q n M d ( n ) ( C ∗ r ( G )) naturally . Since C ∗ r ( H ) embeds in to Q n M d ( n ) ( C ) L n M d ( n ) ( C ) in a wa y that can b e lifted to a sequence of group homomorphisms, this implies C ∗ r ( G × H ) em b eds into Q n M d ( n ) ( C ∗ r ( G )) L n M d ( n ) ( C ∗ r ( G )) in a wa y that can b e lifted to a sequence of group homomorphisms. Now, apply the assumption that G is PMF to obtain the result. Item (2) for PFF follows by noting that, in the ab ov e, the finite-dimensional rep- resen tations obtained that strongly conv erge to G × H is a sequence of tensor represen tations. More precisely , if σ m : G → U ( d 1 ( m )) and φ k : H → U ( d 2 ( k )) con verge strongly to G and H , resp ectively , then there exist sequences of natural n umbers m ( n ) and k ( n ) s.t. σ m ( n ) ⊗ φ k ( n ) : G × H → U ( d 1 ( m ( n )) d 2 ( k ( n ))) con- v erges strongly to G × H . Note that if σ m and φ k factor through finite quotients, so do es σ m ⊗ φ k . The result follows. □ The following is elementary . Lemma 2.5. L et G b e a finite gr oup and H < G . Then G ∗ H ( H × Z ) is virtual ly fr e e and thus in p articular PFF. Pr o of. Define a homomorphism π : G ∗ H ( H × Z ) → G given by the identit y map on G and sending Z to 1. Note that the kernel of π intersected with any conjugate of G is the trivial group. Thus, the action of ker( π ) on the Bass–Serre tree gives a splitting as a graph of groups with trivial edge groups and with some trivial vertex groups and some Z vertex groups [Ser80]. In particular it must actually b e a free group. Hence G is virtually free, which implies PFF by Lemma 2.4. □ No w w e can prov e our main theorem: Pr o of of The or em 1.1. W e first prov e the case L = Z . Note that since the K i are finite index in G , w e ha v e G/K i ∗ H i /K i ( H i /K i × Z ) is PFF by Lemma 2.5. These are exact since amalgamated free pro ducts of exact groups are exact [Dyk04, Theorem 3.2]. Since G is MF (resp., PMF, PFF) by hypothesis, we hav e that [ G/K i ∗ H i /K i ( H i /K i × Z )] × G is also MF (resp., PMF, PFF) by [HS10, Prop 3.2] or Lemma 2.4. No w the result follo ws from Lemma 2.3. No w let L be a general residually finite, MF (resp., PMF, PFF) group. Since either G or L is exact and G is MF (resp., PMF, PFF), G × L is MF (resp., PMF, PFF). Since L is residually finite, { e } is an intersection of finite index subgroups of L , so A NEW SOURCE OF PUREL Y FINITE MA TRICIAL FIELDS 9 H ×{ e } ⊂ G × L is separable. By the previous paragraph, ( G × L ) ∗ H ×{ e } ( H ×{ e }× Z ) is MF (resp., PMF, PFF). Now, note that w e hav e the following embeddings: G ∗ H ( H × L )  → ( G × L ) ∗ H ×{ e } ( G × L )  → ∗ H ×{ e } ( G × L ) ⋊ Z ∼ = ( G × L ) ∗ H ×{ e } ( H × { e } × Z ) . As subgroups of MF (resp., PMF, PFF) groups are MF (resp., PMF, PFF), this concludes the pro of. □ Pr o of of Cor ol lary 1.3. W e pro ceed by induction on the num ber of vertices. The base case of one vertex is clear. Now supp ose any graph pro duct ov er a graph with n − 1 v ertices of exact, MF (resp., PMF, PFF), residually finite groups is MF (resp., PMF, PFF). Let Γ = ( V , E ) b e a graph with n vertices. Pick a v ertex v ∈ V . F or a subset W ⊂ V , let Γ W b e the subgroup of Γ G generated by { G w : w ∈ W } . F or a vertex v ∈ V , let lk( v ) b e the link of v ; that is, the v ertices w ∈ V such that ( v , w ) ∈ E . Recall that Γ G = Γ V \{ v } ∗ Γ lk( v ) (Γ lk( v ) × G v ). T o apply Theorem 1.1, all of the conditions are immediately satisfied by the h yp otheses except that Γ lk( v ) is separable. But this follows from the residual finiteness of all of the G w , see Lemma 3.9 of [HW99]. So the graph pro duct Γ G is MF (resp., PMF, PFF). □ F or the b enefit of the reader we also include the follo wing pro of. Remark 2.6. Limit groups are PFF. Pr o of. W e first recall that all limit groups embed in to an iterated extension of cen tralizers of a free group; i.e., if G is a limit group then one can embed G into a group G n obtained by starting with a free group G 0 and forming amalgamated free pro ducts of the form G i +1 = G i ∗ A i ( A i × Z ) for some cyclic subgroups A i . See for instance Theorem 4.2 of [CG05]. All such G i are LERF groups [Wil08], so in particular each A i is separable. As free groups are PFF, the result follo ws by recursiv ely applying Theorem 1.1. □ References [AGKEP25] T attw amasi Amrutam, David Gao, Sriv atsav Kunnaw alk am Ela y av alli, and Gregory Patc hell, Strict c omparison in re duc e d gr oup C ∗ -algebr as , Inv en t. Math 242 (2025), 639–657. [Ago13] Ian Agol, The virtual Haken c onje cture , Do c. Math. 18 (2013), 1045–1087, With an appendix b y Ian Agol, Daniel Grov es, and Jason Manning. 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Dep ar tment of Ma thema tical Sciences, UCSD, 9500 Gilman Dr, La Jolla, CA 92092, USA Email addr ess : weg002@ucsd.edu URL : https://sites.google.com/ucsd.edu/david-gao Dep ar tment of Ma th ema tics, UMD, Kir w an Hall, Campus Drive, MD 20770, USA Email addr ess : sriva@umd.edu URL : https://sites.google.com/view/srivatsavke/home Dep ar tment of Ma th ema tics, University of W a terloo, W a terloo, Ont ario, Canad a Email addr ess : amanzoor@waterloo.edu URL : https://aareyanmanzoor.github.io Ma thema tical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Obser v a tor y Quar ter, Woodstock R oad, Oxford, OX2 6GG, UK Email addr ess : greg.patchell@maths.ox.ac.uk URL : https://sites.google.com/view/gpatchel