#### 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.

Notes on file formats
Koji Fujiwara, Takashi Shioya, Saeko Yamagata

Mathematics Institute, Tohoku University, Sendai 980-8578, Japan

Email: fujiwara@math.tohoku.ac.jp, shioya@math.tohoku.ac.jp, sa1m28@math.tohoku.ac.jp

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