On homological coherence of discrete groups

By Gunnar E. Carlsson, Boris Goldfarb. 2004

51 citations recorded by the source corpus.

Abstract 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 finite-dimensional regular ring. This class contains word-hyperbolic groups, Coxeter groups and, as we show, the cocompact discrete subgroups of connected Lie groups.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/09755ee9b51177691df5de963cdf59f99605e08c

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The integral K-theoretic Novikov conjecture for groups with finite asymptotic dimension (2004, result): “We proved in [ 5 ] that cocompact lattices in connected Lie groups also have finite asymptotic dimension.”
Asymptotic Dimension (2007, background): “[29] For a connected Lie group G and its maximal compact subgroup K there is a formula asdimG/K = dimG/K where G/K is endowed with a G-invariant metric.”
The asymptotic dimension of a curve graph is finite (2005, background): “First, since Sp(4, Z) ⊂ GL(4, Z), we have asdimSp(4, Z) ≤ asdim GL(4, Z) ≤ 9 by [12], where 9 is the dimension of the symmetric space associated to SL(4, R).”