Algebraic and Geometric Topology 4 (2004), paper no. 38, pages 861-892.

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

Koji Fujiwara, Takashi Shioya, Saeko Yamagata

Abstract. 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.

Keywords. CAT(0) space, parabolic isometry, Artin group, Heisenberg group, geometric dimension, cohomological dimension

AMS subject classification. Primary: 20F67. Secondary: 20F65, 20F36, 57M20, 53C23.

DOI: 10.2140/agt.2004.4.861

E-print: arXiv:math.GT/0308274

Submitted: 17 September 2003. (Revised: 30 July 2004.) Accepted: 13 September 2004. Published: 9 October 2004.

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Koji Fujiwara, Takashi Shioya, Saeko Yamagata
Mathematics Institute, Tohoku University, Sendai 980-8578, Japan

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