Finite group extensions and the Baum-Connes conjecture

Thomas Schick^{∗}

Last compiled October 5, 2000; last edited Sep 22, 2000 or later

Abstract

In this note, we exhibit a method to prove the Baum-Connes conjecture (with coefficients) for extensions with finite quotients of certain groups which already satisfy the Baum-Connes conjecture. Interesting examples to which this method applies are torsion-free finite extensions of the pure braid groups, e.g. the full braid groups.

The Baum-Connes conjecture (in this note always with coefficients) for a group G (all groups in this note are assumed to be discrete and countable) states that the Baum-Connes map

μ_{r}: K_{∗}^{G}(\underline{E}G, A) → K_{∗}(C_{red}^{∗}(G, A))  (1)

is an isomorphism for every C^{∗}-algebra A with an action of G by C^{∗}-algebra homomorphisms.

Here, the left hand side is the equivariant K-homology with coefficients in A of the universal space \underline{E}G for proper G-actions, which is homological in nature. The right hand side, the K-theory of the reduced crossed product of A and G, belongs to the world of C^{∗}-algebras and —to some extend— representations of groups. If A = ℂ with the trivial action, the right hand side becomes the K-theory of the reduced C^{∗}-algebra of G. If, in addition, G is torsion-free, the left hand side is the K-homology of the classifying space of G.

The Baum-Connes conjecture has many important connections to other questions and areas of mathematics. The injectivity of μ_{r} implies the Novikov conjecture about homotopy invariance of higher signatures. It also implies the stable Gromov-Lawson-Rosenberg conjecture about the existence of metrics with positive scalar curvature on spin-manifolds. The surjectivity, on the other hand, gives information in particular about C_{red}^{∗}G. If G is torsion-free, it implies e.g. that this C^{∗}-algebra contains no idempotents different from zero and one. Since we are only considering the Baum-Connes conjecture with coefficients, all these properties follow for all subgroups of G, as well.

We do not want to repeat the construction of the K-groups and the map in the Baum-Connes conjecture (1), instead, the reader is referred to [1, 6]. Recently, Gromov and others have produced counterexamples to the conjecture (with non-trivial, commutative coefficients). However, we will concentrate on groups for which the Baum-Connes conjecture is known to be true and will prove it for new examples. One of the stunning facts is that the Baum-Connes conjecture is inherited by arbitrary subgroups, and there are rather precise results describing its behavior under group extensions (we will recall this below). However, it is almost completely unknown what is happening for an extension with finite quotient of a group which satisfies the Baum-Connes conjecture. The most prominent open examples are probably the full braid groups, which contain the pure braid groups as subgroups of finite index. For the latter, the Baum-Connes conjecture is well known to be true.

∗e-mail: thomas.schick@math.uni-muenster.de www: http://wwwmath.uni-muenster.de/u/lueck/ Fax: ++49 -251/83 38370

The main goal of this note is to prove Baum-Connes for the full braid groups, and for other classes of groups which arise as (finite) extensions of groups for which Baum-Connes is known. We use the following results:

2. Theorem. [9] Assume G is a group acting on a tree. The Baum-Connes conjecture is true for G if and only if it is true for every isotropy subgroup of the action on the vertices.

This implies in particular:

3. Theorem. Assume G is a group which satisfies the Baum-Connes conjecture. If H is a subgroup of G, then H satisfies the Baum-Connes conjecture, too.

4. Theorem. ([2],[8, Theorem 3.1] Assume we have an extension of groups

1 → H → G ⟶^{π} Q → 1.

For every finite subgroup E of Q let H_{E} < G be the inverse image under π of E in G. Assume Q and all groups H_{E} satisfy the Baum-Connes conjecture.

Then the Baum-Connes conjecture is also true for G.

5. Theorem. [4, 5] (Compare also [6]): Let G be an amenable group. Then the Baum-Connes conjecture holds for G.

6. Corollary. Assume we have an extension 1 → H → G → A → 1 of groups, A is torsion-free and amenable, and the Baum-Connes conjecture holds for H. Then it also holds for G.

Our (naive) idea to deal with finite group extensions is to exhibit them as extension with torsion-free quotient. More precisely, assume we have an extension 1 → H → G → Q → 1 where H satisfies the Baum-Connes conjecture and Q is a finite group. Our goal is, to find a normal subgroup U of G which is contained in H and such that G/U is torsion-free and amenable (e.g. virtually nilpotent or virtually solvable). If H is torsion-free, a necessary condition for the existence of such a factorization is that G is torsion-free, too. We will observe that there are large classes of torsion-free groups such that every torsion-free finite extension admits a factorization of the type we are looking for, including all pure braid groups.

We need the following notation:

7. Definition. Let H be a group and p a prime number. Let Ĥ^{p} denote the pro-p completion of H, i.e. the inverse limit of the system of finite p-group quotients of H. There is a natural homomorphism H → Ĥ^{p} (not necessarily injective). The Galois cohomology H^{∗}(Ĥ^{p}, ℤ/p) is defined to be the direct limit of the cohomology of the finite p-group quotients of H (with ℤ/p-coefficients). For the theory of profinite groups and Galois cohomology compare e.g. [11].

The group H is called cohomologically complete if the natural homomorphisms H^{∗}(Ĥ^{p}, ℤ/p) → H^{∗}(H, ℤ/p) are isomorphisms for every prime p.

The second property of H we are going to use is the existence of many quotients which are torsion-free and amenable.

8. Definition. Let H be a group. We say H has enough torsion-free quotients if for every normal subgroup U of H of finite p-power index (for some prime p) another normal subgroup V of H exists which is contained in U and such that H/V is torsion-free and elementary amenable. We say H has enough nilpotent torsion-free quotients if we find V such that H/V is nilpotent.

9. Remark. Recall that the class of elementary amenable groups is the smallest class of groups which contains all finite and all abelian groups and which is closed under extensions and directed unions. It contains in particular all nilpotent and all solvable-by-finite groups.

For the purpose of our paper, we could weaken the condition to H/V being amenable. However, this would deviate from the notation used in [7]. Moreover, all examples known to us satisfy the condition that H/V is solvable-by-finite.

These two properties are important because of the following result:

10. Theorem. [7, Theorem 3.46]: Let H be a group which is cohomologically complete and which has enough torsion-free quotients. Assume that there is a finite model for the classifying space of H.

Let 1 → H → G → Q → 1 be an extension with finite quotient Q and such that G is torsion-free. Then there is a normal subgroup U of G, contained in H, such that G/U is torsion-free and elementary amenable.

The proof of this theorem uses Sylow’s theorems to reduce to the case where Q is a p-group. One then uses the pro-p completions as an intermediate step to show that if no such torsion-free quotient exists, then the projection G → Q induces a split injective map in cohomology (we might have to replace Q with a non-trivial subgroup first). Using an Atiyah-Hirzebruch spectral sequence argument, we show that the same is true for stable cohomotopy. But then a fixed-point theorem of Jackowski implies that the map G → Q itself splits, which contradicts the assumption that G is torsion-free.

We now obtain our main technical result:

11. Theorem. Let H be a group with finite classifying space which is cohomologically complete and which has enough torsion-free quotients. Assume H satisfies the Baum-Connes conjecture. Let 1 → H → G → Q → 1 be an extension of groups such that G is torsion-free and such that Q satisfies the Baum-Connes conjecture. Then the same is true for G.

Proof. Because of Theorem 4 it suffices to prove the result if Q is finite (and still G torsion-free). In this case, by Theorem 10 we can replace the extension 1 → H → G → Q → 1 by another extension 1 → U → G → A → 1 where U is a subgroup of H and A is torsion-free and elementary amenable. Theorem 3 implies that U fulfills the Baum-Connes conjecture. Because of Corollary 6 the same is then true for G.

In the last part of this note, we discuss examples of groups H, to which Theorem 11 applies.

12. Definition. Let F be the class of groups G which fulfill the following properties: G has a finite classifying space, G is cohomologically complete, and G has enough nilpotent torsion-free quotients.

Let U_{tf} be the smallest class of countable group which contains all torsionfree, elementary amenable groups, and which is closed under extensions, directed unions, amalgamated free products, and HNN-extensions.

Set F^{+} := F ∩ U_{tf}.

13. Proposition. If H ∈ U_{tf} then the Baum-Connes conjecture is true for H.

Proof. Note first that all groups in U_{tf} are torsion-free. By Theorem 5 all amenable groups satisfy the Baum-Connes conjecture. Because of Theorem 4 this property is stable under extensions with torsion-free quotients. It follows from Bass-Serre theory [10] that directed unions, amalgamated free products and HNN-extensions act on trees such that the isotropy subgroups are isomorphic to the groups involved in the construction. Hence, by Theorem 2 the Baum-Connes conjecture is also stable under these constructions, and consequently holds for all groups in U_{tf}.

14. Corollary. Assume H ∈ F^{+} and we have an extension

1 → H → G → Q → 1

where Q fulfills the Baum-Connes conjecture and G is torsion-free. Then the Baum-Connes conjecture is true for G.

15. Definition. A semidirect product H ⋊ Q of two groups H and Q is called H_{1}-trivial if the induced action of Q on H_{1}(H, ℤ) is trivial.

16. Definition. A one-relator group G is called primitive if it is finitely generated and if it has a presentation G = ⟨x_{1}, … x_{d} | r⟩ such that the element r in the free group F generated by x_{1}, …, x_{d} is contained in the lower central series subgroup γ_{n}(F) but not in γ_{n+1}(F) and the image of r in γ_{n}(F)/γ_{n+1}(F) is not a proper power.

17. Example. Fundamental groups of orientable two-dimensional surfaces are primitive one-relator groups, as well as one-relator groups where the least common multiple of the exponent-sums for the different generators in the relator is one.

18. Definition. We denote the fundamental group G of the complement of a tame link with d components in S^{3} a link group with d components. We define the linking diagram to be the edge-labeled graph whose vertices are the components of the link, and such that any pair of vertices is joined by exactly one edge. Each edge is labeled with the linking number of the two link components involved.

We say the link group G is primitive if for each prime p there is a spanning subtree of the linking diagram such that none of the labels of the edges of this subtree is congruent to 0 modulo p.

Observe that in particular every knot group (i.e. a link group with only one component) is primitive.

19. Theorem. [7, Theorem 4.26 and Corollary 4.27] The class F^{+} is closed under H_{1}-trivial semidirect products. It contains

• all primitive link groups

• all primitive one-relator groups

• the fundamental groups of all fiber-type arrangements (as defined in [3])

• Artin’s pure braid groups P_{n}

20. Corollary. All the torsion-free finite extensions of the groups listed in Theorem 19, in particular Artin’s full braid groups, satisfy the Baum-Connes conjecture.

In Corollary 14 we can relax the condition that H belongs to F^{+} a little bit:

21. Proposition. Assume G_{1}, …, G_{N} ∈ F^{+}, and Q ∈ U_{tf} has a finite classifying space, is cohomologically complete, and has enough torsion-free quotient. For example, Q could be a free product Q = Q_{1} ∗⋯∗Q_{m} of finitely many groups Q_{i} ∈ F^{+}. Define

G := G_{1} ⋊ (G_{2} ⋊ (⋯ (G_{N} ⋊ Q))),

where each semidirect product is H_{1}-trivial. Then G has a finite classifying space, enough torsion-free quotients, and is cohomologically complete. If there is an exact sequence

1 → G → H → A → 1

with H torsion-free such that A satisfies the Baum-Connes conjecture, then the same is true for H.

Proof. In [7, Proposition 4.30] we prove that G has the desired properties. Moreover, it follows as in the proof of Proposition 13 that G fulfills the Baum-Connes conjecture. The last statement now follows from Theorem 11.

The author thanks Marco Varisco for carefully proofreading the manuscript.

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6 Thomas Schick

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