Equivariant covers for hyperbolic groups. (English) Zbl 1185.20045

Bartels, Arthur; Lück, Wolfgang; Reich, Holger

Geom. Topol. 12, No. 3, 1799-1882 (2008).

M. Gromov [in Geometric group theory. Volume 2: Asymptotic invariants of infinite groups. Lond. Math. Soc. Lect. Note Ser. 182. Cambridge: Cambridge University Press (1993; Zbl 0841.20039)] defined the notion of asymptotic dimension of a metric space \(X\), as the smallest number \(N\) such that for every \(\alpha>0\) there exists an open cover \(\mathcal U\) of \(X\) with the following three properties: (i) \(\dim(\mathcal U)\leq N\); (ii) the Lebesgue number of \(\mathcal U\) is at least \(\alpha\), i.e., for every \(x\in X\), there exists \(U\in\mathcal U\) such that the open ball of radius \(\alpha\) around \(x\) is contained in \(U\); (iii) the elements of \(\mathcal U\) have uniformly bounded diameters.
Gromov also showed in that paper that hyperbolic groups have finite asymptotic dimension.
The main result of the paper under review is an equivariant version of that result. The result is used in the proof the same authors gave of the Farrell-Jones conjecture for \(K_*(RG)\) for every word-hyperbolic group \(G\) and every coefficient ring \(R\) [A. Bartels, W. Lück, and H. Reich, Invent. Math. 172, No. 1, 29-70 (2008; Zbl 1143.19003)].

Reviewer: Athanase Papadopoulos (Strasbourg)

MSC:

20F67	Hyperbolic groups and nonpositively curved groups
20F65	Geometric group theory
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
57M07	Topological methods in group theory

Keywords:

asymptotic dimension; word-hyperbolic groups

Citations:

Zbl 0841.20039; Zbl 1143.19003

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