Parabolic isometries of CAT(0) spaces and CAT(0) dimensions

Details Parabolic isometries of CAT(0) spaces and CAT(0) dimensions Parabolic isometries of CAT(0) spaces and CAT(0) dimensions

2003-08-28

arxiv.org/abs/math/0308274v3

Algebr. Geom. Topol. 4 (2004) 861-891

Mathematics

Geometric Topology

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-38.abs.html

Scientific paper

We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady-Crisp and Bridson, we show that there exist finitely presented groups of geometric dimension 2 which do not act properly on any proper CAT(0) spaces of dimension 2 by isometries, although such actions exist on CAT(0) spaces of dimension 3. Another example is the fundamental group, G, of a complete, non-compact, complex hyperbolic manifold M with finite volume, of complex-dimension n > 1. The group G is acting on the universal cover of M, which is isometric to H^n_C. It is a CAT(-1) space of dimension 2n. The geometric dimension of G is 2n-1. We show that G does not act on any proper CAT(0) space of dimension 2n-1 properly by isometries. We also discuss the fundamental groups of a torus bundle over a circle, and solvable Baumslag-Solitar groups.

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