On Bestvina-Mess Formula

2005/03/01 by A. N. Dranishnikov, Dranishnikov, A. N.
Mathematics · #20F55 #55M10 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR) #math.AT #math.GR #msc:20F55 #msc:55M10

paper · pdf · doi:10.48550/arxiv.math/0503018

10 pages

arxiv created 2005/03/01 · arxiv updated 2009/12/01

Abstract

Bestvina and Mess [BM] proved a remarkable formula for torsion free hyperbolic groups dim_L∂Γ=cd_LΓ-1 connecting the cohomological dimension of a group Γ with the cohomological dimension of its boundary ∂Γ. In [Be] Bestvina introduced a notion of \sZ-structure on a discrete group and noticed that his formula holds true for all torsion free groups with \sZ-structure. Bestvina's notion of \sZ-structure can be extended to groups containing torsion by replacing the covering space action in the definition by the geometric action. Though the Bestvina-Mess formula trivially is not valid for groups with torsion, we show that it still holds in the following modified form: \it The cohomological dimension of a \sZ-boundary of a group Γ equals its global cohomological dimension for every PID L as the coefficient group dim_L∂Γ=gcd_L(∂Γ). Using this formula we show that the cohomological dimension of the boundary dim_L∂Γ is a quasi-isometry invariant of a group.

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