#### Volume 4, issue 2 (2004)

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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
 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 $CAT\left(0\right)$ spaces of dimension $2$ by isometries, although such actions exist on $CAT\left(0\right)$ 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 $n\ge 2$. The group $G$ is acting on the universal cover of $M$, which is isometric to ${H}_{ℂ}^{n}$. It is a $CAT\left(-1\right)$ space of dimension $2n$. The geometric dimension of $G$ is $2n-1$. We show that $G$ does not act on any proper $CAT\left(0\right)$ 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
##### Mathematical Subject Classification 2000
Primary: 20F67
Secondary: 20F65, 20F36, 57M20, 53C23
##### 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