On the K-theory of groups with finite asymptotic dimension

Abstract

Abstract. It is proved that the assembly maps in algebraic K-and L-theory with respect to the family of finite subgroups is injective for groups Γ with finite asymptotic dimension that admit a finite model for EΓ. The result also applies to certain groups that admit only a finite dimensional model for EΓ. In particular, it applies to discrete subgroups of virtually connected Lie groups.

Amenable covers, volume and L 2-Betti numbers of aspherical manifolds

“…Such a representing cycle of the fundamental class leads to an estimate for b (2) i ( M ) for all i ≥ 0 (in our case: b (2) i ( M ) ≤ const C,n vol(M )) by a Poincaré duality argument. See Theorem A.1 in the Appendix.…”

“…Notice that for aspherical M we have b (2) i ( M ) = b (2) i (π 1 (M )), and b (2) i (π 1 (M )) is an orbit equivalence invariant of the fundamental group π 1 (M ) by Gaboriau's work [17]. It is a particularly interesting aspect of the previous corollary that it provides a non-trivial orbit equivalence invariant that bounds the minimal volume from below.…”

“…For each n ∈ N there is a constant const n > 0 with the following property: If M is an n-dimensional, closed, aspherical manifold M , then b (2) i ( M ) ≤ const n minvol(M ) for all i ≥ 0. These two corollaries are the analog of Gromov's main inequality [19, 0.5] with simplicial volume replaced by L 2 -Betti numbers and the additional hypothesis of asphericity.…”

“…L 2 -Betti numbers for regular coverings of closed Riemannian manifolds were introduced by Atiyah [1], and their range of definition and application was widened over the years [7,[10][11][12]26]. By now there is a definition of L 2 -Betti numbers b (2) i (Y ; Γ) ∈ [0, ∞], i ≥ 0, of an arbitrary space Y with the action of a discrete group Γ [27, chapter 6]. The most important case for us is the universal covering M of a space M with the natural action of its fundamental group π 1 (M ).…”

“…For each n ∈ N there is a constant const n > 0 with the following property: If M is an n-dimensional, closed, aspherical Riemannian manifold with lower Ricci curvature bound Ricci(M ) ≥ −(n − 1), then b (2) i ( M ) ≤ const n vol(M ) for all i ≥ 0.…”