Weak coherence of groups and finite decomposition complexity

Abstract

The weak regular coherence is a coarse property of a finitely generated group Γ. It was introduced by G. Carlsson and this author to play the role of a weakening of Waldhausen's regular coherence as part of computation of the integral K-theoretic assembly map.A new class of metric spaces (sFDC) was introduced recently by A. Dranishnikov and M. Zarichnyi. This class includes most notably the spaces with finite decomposition complexity (FDC) studied by E. Guentner, D. Ramras, R. Tessera, and G. Yu. The main theorem of this paper shows that a group that has finite K(Γ, 1) and sFDC is weakly regular coherent.As a consequence, the integral K-theoretic assembly maps are isomorphisms in all dimensions for any group that has finite K(Γ, 1) and FDC. In particular, the Whitehead group Wh(Γ) is trivial for such groups.

Colimit theorems for coarse coherence with applications

“…The fact that groups of finite asymptotic dimension are coarsely coherent was shown in [1]. This was generalised to groups of countable asymptotic dimension in [5]. The class of coarsely coherent groups has lots of permanence properties, such as invariance under very general extensions, products, etc.…”

“…Choose k = 2D + 2b + 2d. Since we can write Z ∞ as the union of (2D + 2b + 2d)-disjoint sets, it follows from the main theorem of [5] that…”

Topological rigidity of higher graph manifolds

“…A group satisfying these two conditions has been proven to also satisfy the integral isomorphism conjecture in algebraic K-theory, according to Theorem 3.11 of Goldfarb in [12].…”

“…On the other hand, relationships between several generalizations of the concept of finite asymptotic dimension in connection with isomorphism conjectures, in algebraic K and Ltheory , as well as coarse versions of these have been carried out by G. Carlsson and B. Goldfarb in [7], [12], [8].…”

Hyperbolic Dimension and Decomposition Complexity

“…These results were obtained by studying certain assembly maps in L-theory and topological K-theory. The assembly map in algebraic K-theory has been studied for groups with FDC by several authors, including Ramras, Tessera and Yu [RTY14], Kasprowski [Kas15], and Goldfarb [Gol13].…”