On homological coherence of discrete groups

On Homological Coherence of Discrete Groups

We explore a weakening of the coherence property of discrete groups studied by F. Waldhausen. The new notion is defined in terms of the coarse geometry of groups and should be as useful for computing their K-theory. We prove that a group Γ of finite asymptotic dimension is weakly coherent. In particular, there is a large collection of R[Γ]-modules of finite homological dimension when R is a fin...

Homological Infiniteness of Torelli Groups

1 → I g,r → M n g,r → Sp(2g,Z) → 1. We omit the decorations n and r when they are zero. In a series of papers [7, 8, 9, 10], D. Johnson obtained several fundamental results concerning the structure of Ig and Ig,1 (see also [11, 2]). In particular, he proved that Ig and Ig,1 are finitely generated for all g ≥ 3. On the contrary, A. Miller and D. McCullough [14] showed that I2 (and hence I n 2,r ...

Homological Stability for Automorphism Groups

We prove a general homological stability theorem for families of automorphism groups in certain categories. We show that this theorem can be applied to all the classical examples of stable families of groups, such as the symmetric groups, general linear groups and mapping class groups, and obtain new stability theorems with twisted coefficients for the braid groups, automorphisms of free groups...

Structural Properties of Inner Amenable Discrete Groups

Homological Stability for automorphism groups of Raags

The stabilization of Hi(Aut(G);Z) for i large has been shown to hold for most groups by the main theorem of [3] and [9, Corollary 1.3].1 The stabilization of Hi(Aut(G);Z) in contrast has so far only been known in two extreme cases: when G is abelian and when G has trivial center and does not factorize as a direct product. Indeed, in the first case Aut(Gn) is isomorphic to GLn(End(G)), which is ...