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

Fujiwara, Koji; Shioya, Takashi; Yamagata, Saeko

doi:10.2140/agt.2004.4.861

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

Koji Fujiwara, Takashi Shioya and Saeko Yamagata

Algebraic & Geometric Topology 4 (2004) 861–892

DOI: 10.2140/agt.2004.4.861

arXiv: math.GT/0308274

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 $n \geq 2$ . The group $G$ is acting on the universal cover of $M$ , which is isometric to $H_{ℂ}^{n}$ . It is a $CAT (- 1)$ space of dimension $2 n$ . The geometric dimension of $G$ is $2 n - 1$ . We show that $G$ does not act on any proper $CAT (0)$ space of dimension $2 n - 1$ 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

Mathematical Subject Classification 2000

Primary: 20F67

Secondary: 20F65, 20F36, 57M20, 53C23

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Publication

Received: 17 September 2003

Revised: 30 July 2004

Accepted: 13 September 2004

Published: 9 October 2004

Authors

Koji Fujiwara
Mathematics Institute Tohoku University Sendai 980-8578 Japan

Takashi Shioya
Mathematics Institute Tohoku University Sendai 980-8578 Japan

Saeko Yamagata
Mathematics Institute Tohoku University Sendai 980-8578 Japan

Volume 4, issue 2 (2004)