Idempotent Matrices Over Group Rings

Transcription

1 Panjab University Chandigarh Indian Institute of Science Education and Research Mohali Georg-August Universität Göttingen 25 June 2012

2 Idempotent matrices over group rings Abstract I will give a survey of some of the work on the Bass conjecture for the Hattori-Stallings ranks of finitely generated projective modules over group rings R[G], or equivalently for the traces of idempotents in matrix rings M n (R[G]) over R[G]. The focus will mainly be on the approach to investigate this conjecture via the cyclic homology of group algebras.

3 Let G be a (multiplicative) group and let R be a commutative ring with identity. The group ring R[G] of G over R is the ring whose elements are the finite formal sums g G,α(g) R α(g)g, α(g) = 0 for all but finitely many g G, with addition defined coefficient-wise and multiplication defined distributively using the multiplication in G: α(g)g + β(g)g = (α(g) + β(g))g, ( ) α(g)g. β(h)h = α(x)β(y) g. xy=g Our aim here is to give a survey of some of the work on Bass conjecture ([Bas76], [Bas79]) for the Hattori-Stallings ranks of finitely generated projective modules over R[G], or equivalently for the traces of idempotents in matrix rings M n (R[G]) over R[G].

4 A number of positive results are now known; the conjecture, however, still remains unresolved. Our focus in this talk will mainly be on the approach to investigate this conjecture via the cyclic homology of group algebras.

5 Trace maps Let k be a commutative ring and A a k-algebra, and M a k-module. A k-linear map τ : A M is called a trace map if τ(ab) = τ(ba) for all a, b A. Let A [2] := [A, A] be the k-submodule of A spanned by all elements of the type ab ba with a, b A, Universal trace map: The canonical projection τ : A A/A [2], a A, a a + A [2], is the universal trace map on A.

6 Higher trace maps Let θ : R A be a k-algebra epimorphism with ker I. For n 1, a k-homomorphism R M which vanishes on [R, R] + I n is called a higher trace map of degree n on A relative to the extension 0 I R A 0. Universal higher trace map of degree n: The canonical projection τ n : R R/(I n + R [2] ) is the universal higher trace of degree n relative to the extension 0 I R A 0.

7 Example Let R[G] be the group ring of a group G over a commutative ring R with identity and let M n (R[G]) be the ring of n n matrices over R[G]. For every conjugacy class C of G, there is a trace map ɛ C : R[G] R defined by α(g)g = α(g) g C ɛ C g G, α(g) R which extends to a trace map ɛ C : M n (R[G]) R by setting ɛ C (A) = ɛ C (Tr(A)), A M n (R[G]), where Tr(A) is the usual trace of the matrix A.

8 Remark The above trace functions provide a useful tool for the study of algebraic elements in group algebras. For example, if ξ is an element of the complex group algebra C[G] satisfying f (ξ) = 0 for some nonzero polynomial f (X ) C[X ], and if λ denotes the maximum of the absolute values of the complex roots of f (X ), then ɛ C (ξ)ɛ C (ξ)/ C λ 2 C where C is the cardinality of the conjugacy class C and denotes complex conjugation [PP90]. A similar result, in fact, holds for algebraic elements in M n (C[G]) for all n 1 [Ale93].

9 Hattori-Stallings rank of a f.g. projective module Let R be a ring with identity and P a finitely generated projective right R-module. Set P = Hom R (P, R) and observe that it carries a left R-module structure: (rf )(x) = rf (x), r R, f P, x P. We then have an isomorphism α : P R P End R (P) given by x f (y xf (y)), y P, x P, f P. Also we have a homomorphism t : P R P R/R [2], x f f (x) + R [2], x P, f P, where R [2] denotes the additive subgroup of R generated by all elements of the type rs sr (r, s R).

10 Thus, for a finitely generated projective right R-module P, we have a homomorphism T P : End R (P) R/R [2] given by T P = t α 1. The element T P (1 P ), denoted r P, is called the Hattori-Stallings rank of P (Hattori [Hat65], Stallings [Sta65]). If (x i ), (f i ) is a finite coordinate system for P [ i.e., elements x i P, f i P such that every x P can be written as x = i x if i (x)], then it turns out that r P = i f i (x i ) + R [2]. (1)

11 Trace maps on group rings If R is the group ring k[g] of a group G over a commutative ring k with identity, then there is an obvious isomorphism R/R [2] C TG kc, where C TG kc is the free k-module on the set TG of conjugacy classes C of G. For a finitely generated projective k[g]-module P, let r P (C) k denote the coefficient of the conjugacy class C in the image of r P under the above isomorphism.

12 Higher traces on group rings Let 1 H G Π 1 be an extension of groups, and k a commutative ring with identity. Let {Hg C } C T Π, gc G be a set of representatives of the conjugacy classes T Π with g {1} = 1. Let B C = {γ k[h] γg C k[g] [2].} Theorem (Mikhailov - Passi [Mik05]) For all n 1, k[g]/( n (H)k[G] + k[g] [2] ) C T Π k[h]/( n (H) + B C ),

13 Swan s Theorem Theorem (Swan [Swa60]) If P is a finitely generated projective module over the integral group ring Z[G] of a finite group G, then Q Z P is a free Q[G]-module. In terms of the Hattori-Stallings rank this means that r P (C) = 0 for all non-identity conjugacy classes C of G. This result of R. G. Swan led H. Bass ([Bas76], [Bas79]) to make the following conjectures:

14 Bass conjectures Let k be a subring of C such that k Q = Z, G an arbitrary group and let P be a finitely generated projective k[g]-module. Conjecture Bass strong conjecture: r P (C) = 0 for all non-identity conjugacy classes C of G. Conjecture Bass weak conjecture: C {1} r P(C) = 0.

15 Matrix version Equivalent versions of these conjectures are the following statements for the trace functions ɛ C on idempotent matrices E over k[g]: Conjecture ɛ C (E) = 0 for all non-identity conjugacy classes C of G. Conjecture C {1} ɛ C (E) = 0.

16 Geoghegan s conjecture If G be a finitely presented group, X a finite connected complex with Π 1 (X, ) G, and f : (X, ) (X, ) a pointed homotopy idempotent inducing identity on the fundamental group, then the Nielsen number N(f ) of f is either zero or one. N(f ):= the number of essential fixed point classes of f.

17 Bass strong conjecture (over Z) is equivalent to Geoghegan s conjecture.

18 R. Geoghegan: Fixed points in finitely dominated compacta: the geometric meaning of a conjecture of H. Bass. In: Shape theory and geometric topology (Dubrovnik, 1981), Lecture Notes in Math. 870, Springer, Berlin (1981) 622. R. Geoghegan, Nielsen fixed point theory. In: Handbook of geometric topology, NorthHolland, Amsterdam (2002) A. J. Berrick, I. Chatterji and G. Mislin: Homotopy idempotents on manifolds and Bass conjectures, Geometry & Topology Monographs 10 (2007) 4162.

19 Notation For x G, let [x], x and C G (x) denote respectively the conjugacy class of x, the subgroup generated by x and the centralizer of x in the group G and denote by N x the quotient C G (x)/ x. In the set TG of conjugacy classes of the group G, let G fin (resp: G ) denote the set of conjugacy classes of the elements of finite (resp: infinite) order in G.

20 Torsion conjugacy classes We first consider Bass conjecture for the conjugacy classes of elements of finite order where the strong conjecture is known to hold.

21 Torsion conjugacy classes Theorem (Linnell [Lin83], Moody [Moo86], Schafer [Sch91], Schick [Sch]) Let R be an integral domain, G a group and E M n (R[G]) an idempotent matrix. Let p be any prime not invertible in R and let x G be such that ɛ [x] (E) 0. Then there exist natural numbers f and u such that x x pfu and ɛ [x] (E) = ɛ [x p f ] (E). In particular, ɛ [x](e) = 0 for all non-identity elements x G of finite order not invertible in R.

22 To indicate a proof of this result, we need the following lemmas.

23 Lemma 1 Lemma Let R be a commutative ring of characteristic p r, r 1, G an arbitrary group and E M n (R[G]) an idempotent matrix. Then there exist elements a 1,..., a n R, x 1...., x n G and an integer N > 0 such that, for all s 1, Tr(E) N i=1 a ps i x ps i mod [R[G], R[G]].

24 Lemma 2 Lemma Let m be a maximal ideal in a commutative ring R such that R/m is a finite field. Then, for all a R\m and k 1, (a pk 1 ) pf 1 1 mod m k, where p f is the order of R/m.

25 To prove the first Lemma refer to ([Cli80], Lemmas 1 and 4), and for the second Lemma induct of k 1.

26 Torsion conjugacy classes We can clearly assume that R is finitely generated. Let m be a maximal ideal in R containing p. Then R/m is a finite field of order, say, p f (Wehrfritz [Weh73], 4.1). Since R is a Nötherian domain, k m k = (0). Choose k 0 > 0 such that ɛ [g] (Tr(E)) / m k 0 for every conjugacy class [g] in the finite set S = {[g] : ɛ [g] (Tr(E)) 0}. Let k > k 0 be fixed. Then the characteristic of R/m k is p r with 1 r k. By Lemma 1, there exist a 1,..., a n R, x 1...., x n G and an integer N > 0 such that, for all s 1, Tr(E) N i=1 a ps i x ps i (mod m k + [R[G], R[G]]).

27 Taking s = k 1 and k 1 + f and writing x pk 1 i = y i, a pk 1 i By Lemma 2, = α i, we get modulo m k + [R[G], R[G]], Tr(E) α i y i α pf i y pf i. α pf i = a pk 1+f i a pk 1 i = α i mod m k. Consequently, we have Tr(E) α i y i α i y pf i (mod m k + [R[G], R[G]]). (2) Let β i = y j y i α j and β i = α y pf j y pf j. If [g] S, then (2) i shows that there exists i with y pf i g and β i / m k which in turn implies that there exists j with y pf j ɛ [yj ](E) β j 0 mod m k. y pf i g and

28 Hence the map F : [g] [g pf ] defined on the set of all conjugacy classes in G satisfies that S F (S). Consequently, S being finite, F is bijective on S. Since [x] S, it thus follows that [x] = [x pfu ] for some u > 0; moreover, (2) yields that ɛ [x p f ] (E) ɛ [x](e) mod m k for all k > k 0. Hence ɛ [x p f ] (E) = ɛ [x](e) and the proof is complete.

29 Non-torsion conjugacy classes We next consider some of the approaches to investigate Bass conjecture for the conjugacy classes of elements of infinite order. One such approach is via cyclic homology, first noted by Eckmann [Eck86] and subsequently pursued extensively: [Eckmann [Eck01], Emmanouil ([Emm98b], [Emm98a], [Emm], [Emm01]), Farrell-Linnell [FL03], Emmanouil-Passi ([EP04], [EP06]).]

30 Cyclic homology of algebras Let A be a k-algebra and C the category with objects consisting of all algebra extensions 1 I R A 1 and morphisms the commutative diagrams 1 I R A 1 1 J S A 1 (3)

31 Quillen s characterization of cyclic homology Let k be a field of characteistic zero, A a k-algebra and M k the category of k-modules. For each n 0 the associations of R/(I n + R [2] ) and I n+1 /[I, I n ]) to the extension 1 I R A 1 give functors from the category C to the category M k. HC 2n (A) lim R/(I n+1 +R [2] ), HC 2n+1 (A) lim I n+1 /[I, I n ], n 0.

32 Cyclic homology of group rings Theorem (Burghelea [Bur85]) If R is a commutative ring containing Q, the field of rational numbers, then HC (R[G]) [x] G finh (N x, R) HC (R) [x] G H (N x, R). (See Loday [Lod92], Chapters 7 and 8].)

33 Connes periodicity exact sequence An important operator in cyclic homology of algebras is the Connes periodicity operator S : HC HC 2, and the resulting long exact sequence connecting cyclic homology and Hochschild homology. HH n (A) I HC n (A) S HC n 2 (A) B HH n 1 (A).

34 An application of cyclic homology of group algebras to investigate Bass conjecture results from the following (see Loday [Lod92], Theorem 8.3.4): Theorem (Connes [Con85]) There exist homomorphisms ch 0, n : K 0 (R[G]) HC 2n (R[G]), n 0, such that, for every finitely generated projective R[G]-module P, ch 0, 0 ([P]) = r P and S ch 0, n = ch 0, n 1, where [P] denotes the class in the Grothendieck group K 0 (R[G]) determined by P, and S is the Connes periodicity map.

35 We thus have a homomorphism ch 0, : K 0 (R[G]) lim HC 2n (R[G]). The Connes periodicity map commutes with the Burghelea s decomposition of the cyclic homology groups and so we have, for each element x G of infinite order, an inverse system {S : H 2n (N x, R) H 2n 2 (N x, R)} n 0. Let T ([x]; R) := lim H 2n (N x, R). In view of Connes theorem, we have a homomorphism ch [x] 0, : K 0(R[G]) T ([x]; R).

36 A sufficient condition for Bass conjecture to hold Theorem If T ([x]; R) = 0 for an element x G of infinite order, then the Bass conjecture is satisfied over R for the conjugacy class [x].

37 Burghelea s conjecture Conjecture (Burghelea [Bur85]) If K(G, 1) has the homotopy type of a finite CW-complex, then T ([x]; R) = 0 for all [x] G.

38 For applications of the validity of the Bass conjecture, it is naturally of interest to know when the homomorphism ch 0, : K 0 (R[G]) lim HC 2n (R[G]) is injective.

39 Theorem (Marciniak-Sehgal [Mar96]) If Γ is a finitely generated nilpotent group and k is a characteristic zero splitting field for its torsion subgroup, then ch 0, 0 : K 0 (k[γ ]) HC 0 (k[γ ]) is injective.

40 Centralizers and homological dimension If a group G is such that, for every x G of infinite order, H i (N x, R) = 0 for all sufficiently large i, then clearly lim H 2n(N x, R) = 0, and so G satisfies the Bass conjecture over R for the conjugacy classes in G. With this approach Eckmann [Eck86] proved the following:

41 Theorem (Eckmann [Eck86]) For the groups G of finite homological dimension hd Q G belonging to one of the classes (i) solvable groups of finite Hirsch number, (ii) linear groups in characteristic zero, (iii) groups of cohomological dimension 2 over Q, the Hattori-Stallings rank r P of a finitely generated projective Q[G]-module P vanishes on the conjugacy classes of elements of infinite order.

42 Concerning (ii), it may be noted that, in view of the results of Bass ([Bas76], 9), the strong conjecture, in fact, holds for linear groups. Related to (iii), Chadha-Passi proved the following result: Theorem (Chadha-Passi [Cha94b]) If G is a group of finite cohomological dimension cd k G = n 1 over a field k of characteristic zero with its centre containing a free abelian subgroup of rank n 1, then for any x G of infinite order, the homological dimension hd k C G (x)/ x n 1, and so the Hattori-Stallings rank of a finitely generated projective k[g]-module vanishes on the conjugacy classes in G.

43 The class E(R) With a view to pursue Eckmann s approach further, let us denote by E(R) the class of groups G with homological dimension hd R G < and satisfying hd R (C G (x)/ x ) < for every x G of infinite order. Note that E(R)-groups satisfy the strong Bass conjecture over Z. Closure properties of the class E(R) have been investigated by Chadha-Passi [Cha94a], Ji [Ji,95], and Sykiotis [Syk02].

44 Closure properties of E(R) Theorem The class E(R) is closed under (i) subgroups, (ii) extensions, (iii) free products with amalgamation (iv) HNN extension and (v) directed union of groups with bounded homological dimension over R.

45 Theorem (Ji [Ji,95]). The class E(Q) contains all arithmetic groups.

46 Hyperbolic groups We next consider hyperbolic groups. Let Γ be a finitely generated group with a fixed finite system of generators and let γ denote the length of a shortest word (in these generators) representing given γ Γ. Set (α. β) = 1 2 ( α + β α 1 β ) and call Γ word hyperbolic [Gro87] (or negatively curved) if there exists a constant δ 0 (depending on the choice of generators) such that every three elements α, β, γ Γ satisfy (α.β) min((α.γ), (β.γ)) δ.

47 If Γ is hyperbolic for some finite generating subset G 0 Γ, then Γ is hyperbolic for every finite generating subset G of Γ ([Gro87], Corollary 2.3E). Equivalently, ([Gro87] 7.3, [Ger91]), a finitely generated group is said to be hyperbolic if for some (and hence any) finite generating set A, the Cayley graph Γ A (G) has the property that there is a positive real constant δ such that all geodesic triangles are δ-thin. This means that for any embedding of a triangle in Γ A (G) which is an isometry on each side, a δ-neighborhood of (the image of) any two sides contains the third.

48 Theorem (Ji [Ji,95]). The class E(Q) contains all hyperbolic groups.

49 Theorem (Gersten-Short [Ger91]). Let x be an element of infinite order in the hyperbolic group G. Then C G (x) contains x as a subgroup of finite index. It thus follows that hyperbolic groups satisfy Bass strong conjecture.

50 Growth of groups Let Γ be a finitely generated group and let γ, γ Γ a word length function defined on Γ with respect to a set S = {s 1,..., s n } of generators of Γ. For every non-negative integer m, let C S (m) = {γ Γ : γ m}. The function C S : N N on the set N of non-negative integers is called the growth function of the group Γ associated with the specific choice of generators S. If, for a non-negative integer m, there exists a non-negative constant c such that C S (k) ck m for all k 0, then Γ is said to have polynomial growth of degree m. This definition does not depend on the choice of the particular set S of generators [see Musso and Tricerri [Mus91])].

51 Theorem (Gromov [Gro81]) A finitely generated group has polynomial growth if and only if it is almost nilpotent. From the closure properties of the class E(Q) it follows that All finitely generated groups of polynomial growth belong to the class E(Q).

52 Growth of groups and Bass conjecture A relationship of Bass conjecture with the growth of groups has been examined by Karlsson and Neuhauser [KN04]. Let G be a finitely generated group and S a finite set of generators of G. Karlsson and Neuhauser call an element g G of infinite order a u-element if there exists a constant C > 0 such that the word-length function induced on G by S satisfies g n C log n for all n > 1. It turns out that if an element g G of infinite order is conjugate to g k for some k > 1, then g is a u-element. Thus results the following

53 Theorem (Karlsson-Neuhauser [KN04]) The C[G]-Bass conjecture holds for groups without u-elements.

54 Amenable groups Recall that the class C of elementary amenable groups (Chou [Cho80]) is the smallest class of groups which satisfies the following properties: (i) C contains all abelian groups and all finite groups, and (ii) C is closed under extension and directed union. It follows that the class E(Q) contains the class of elementary amenable groups of finite homological dimension over Q (Chadha-Passi [Cha94a] Corollary 2.2).

55 Farrell and Linnell[FL03] have proved that Bass strong conjecture holds for all elementary amenable groups. It is now known that this is the case for all amenable groups. Theorem (Berrick-Chatterji-Mislin [BCM04]) Amenable groups satisfy the classical Bass conjecture.

56 An obstruction Suppose that R is a finitely generated subring of C such that R Q = Z, G is a finitely generated group, E is an idempotent matrix over R[G] and S is a set of representatives of the conjugacy classes C such that ɛ C (E) 0. Since R is finitely generated, there can only be finitely many primes, say q 1,..., q s which can divide the non-zero partial augmentations ɛ [g] (Tr(E)), g S. Let x S and let p 1, p 2,... be an enumeration of rational primes. Then, for every i 1, there exist an integer a(p i ) > 0 and g i G such that g i xg 1 i = x pa(p i ) i. If r 1, r 2,... is a sequence in which each integer appears infinitely often, then the subgroup H = x, gr gr 1 k xg rk... g r1 k 1 can be seen to be isomorphic to the additive group Q + of rationals and contained in the Inder union Bir S. of Passi the conjugacy Idempotent Matrices classes Over Group of the Ringsfinitely

57 An obstruction Thus: If the Bass conjecture is false for the group ring R[G], then G must contain subgroups H K with K finitely generated and H isomorphic to the additive group of the rationals and contained in the union of only finitely many K-conjugacy classes. The above result is due to Linnell [Lin83] and Moody [Moo87] (see Linnell [Lin93], Theorem 2). It is, however, possible for a group G to contained a pair of subgroup H K as above and still satisfy Bass conjecture.

58 Example (Linnell [Lin93]) There is a 2-generator group G of cohomological dimension 2 containing a subgroup A Q + such that A\1 is contained in a single conjugacy class of G.

59 Emmanouil [Emm98b] has further demonstrated how Linnell s theorem can be used in order to impose certain restrictions on the structure of a potential (residually solvable) counter-example.

60 Theorem (Emmanouil [Emm98b]) Let G be a finitely generated residually solvable group which does not satisfy Bass conjecture. Then, V = i=1 Qv i is a normal subgroup of a certain quotient group of G and the induced Q-linear action of G on V has the following properties: (a) the action factors through a solvable quotient of G, (b) V has no proper non-trivial G-invariant subgroup, (c) v i is contained in the G-orbit of v 1 for all i 1, and (d) the subgroup Qv 1 of V is contained in finitely many G-orbits

61 In [Emm98b] Emmanouil describes an example of Hall [Hal59] of a finitely generated solvable group G, which contains an infinite-dimensional Q-vector space V as a normal subgroup in such a way that the conjugation action of G on V satisfies the conditions (b), (c), and (d) above. Nevertheless, using cyclic homology, this group is seen to satisfy Bass conjecture.

62 Nilpotent cohomology classes Let g be an element of infinite order in the group G. Denote by α(g) the cohomology class in H 2 (N g, Z) corresponding to the central extension 1 Z g C G (g) N g 1. (4) Let C(Q) denote the class of those groups G in which, for every element g of infinite order, the cohomology class α(g) is rationally nilpotent, i.e., its image α(g) Q in the cohomology ring H (N g, Q) is nilpotent. This class has been studied by Emmanouil [Emm98a].

63 It turns out that for G C(Q), the Connes periodicity map S : H (G, Q) H 2 (G, Q) is nilpotent on the components corresponding to each of the conjugacy classes of elements of infinite order and therefore T ([g]; Q) = 0 for g G of infinite order,.consequently: The groups in the class C(Q) satisfy Bass strong conjecture. Moreover, E(Q) C(Q).

64 Theorem (Emmanouil [Emm98a]). Residually C(Q)-groups satisfy Bass strong conjecture.

65 Emmanouil [Emm01] has shown that while the class C(Q) is closed under subgroups, finite direct products and free products, it is not extension closed: there are finitely generated solvable groups which are not in C(Q). [Note, however, that finitely generated metabelian groups are in C(Q).] That the above (cyclic) homological approach to the idempotent conjectures has its limitations has been brought out by Emmanouil by exhibiting examples of groups that are not contained (residually) in C(Q).

66 Bass-Serre theory of groups acting on trees Using the Bass-Serre theory of groups acting on trees, Sykiotis has proved the following results: Theorem (Sykiotis [Syk02]). Let G be the fundamental group of a graph of groups such that all vertex groups are contained in the class C(Q), while all edge groups are torsion. Then G is contained in C(Q); in particular, G satisfies Bass conjecture.

67 Theorem (Sykiotis [Syk02]). Let G be the free product of the groups G λ, λ Λ with amalgamated subgroup H. Suppose that each G λ C(Q) and H is a normal subgroup of G not containing infinitely divisible elements of infinite order (in particular, suppose H is a residually finite normal subgroup of G). Then G satisfies Bass conjecture.

68 The classes A(k) and A (k) (Emmanouil-Passi [EP04]) Let k be a commutative ring. The class A(k) consists of those groups G that satisfy the following condition: For every element g G of infinite order there is a positive integer n such that the cap product map α(g) n : H 2n (N g, k) H 0 (N g, k) k (5) is not surjective. We shall denote the class A(Z) by A.

69 The classes A(k) and A (k) (Emmanouil-Passi [EP04]) The class A (k) consists of those groups G that have the following property: For every element g G of infinite order there exists a field K and a ring homomorphism k K such that the canonical image α(g) K of α ( g) in H 2 (N g, K) is nilpotent in the cohomology ring H (N g, K). The class A (Z) will be denoted by A.

70 C(Q) = A A.

71 Theorem (Emmanouil-Passi [EP04]) If a group G is residually contained in the class A, then it satisfies Bass conjecture.

72 Connes periodicity sequence in low dimensions Connes periodicity sequence in low dimensions gives an exact sequence HC 2 (Z[G]) HC 0 (Z[G]) HH 1 (Z[G]). For an element g G of infinite order, Burghelea s decomposition yields an exact sequence H 2 (N g ) Z[g] H 1 (C(g)) = C(g)/[C(g), C(g)]. It turns out that ker Z[g] H 1 (C(g)) is trivial if g [C(g), C(g)] = {1}. ( ) Consequently, groups whose elements of infinite order have the property ( ) satisfy Bass conjecture [Mik05].

73 Theorem (Guba-Sapir [1999]) Diagram groups have the property ( ).

74 Relaxing the requirement for a group to be in the class A, Panagopoulos [Pan08] considers the class A L consisting of the groups G that satisfy the following condition: If 1 g G is such that there exists 0 u N with the property that g and g nu are conjugate for all n N, then the homomorphism α(g) n : H 2n (N g, Z) H 0 (N g, Z) Z (6) is not surjective.

75 Theorem Panagopoulos [Pan08]) (i) Groups in the class A L satisfy Bass conjecture. (ii) If a group G is residually contained in A L, then it is already in A L.

76 We conclude with the statement of Bass conjecture that is still open. Conjecture ([Bas76], Remark 8.2]) If P is a finitely generated projective C[G]-module, then r P (κ) = 0 for the conjugacy classes κ of elements of infinite order.

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