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2004 Parabolic isometries of CAT(0) spaces and CAT(0) dimensions
Koji Fujiwara, Takashi Shioya, Saeko Yamagata
Algebr. Geom. Topol. 4(2): 861-892 (2004). DOI: 10.2140/agt.2004.4.861


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 n2. The group G is acting on the universal cover of M, which is isometric to Hn. It is a CAT(1) space of dimension 2n. The geometric dimension of G is 2n1. We show that G does not act on any proper CAT(0) space of dimension 2n1 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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Koji Fujiwara. Takashi Shioya. Saeko Yamagata. "Parabolic isometries of CAT(0) spaces and CAT(0) dimensions." Algebr. Geom. Topol. 4 (2) 861 - 892, 2004.


Received: 17 September 2003; Revised: 30 July 2004; Accepted: 13 September 2004; Published: 2004
First available in Project Euclid: 21 December 2017

zbMATH: 1073.20035
MathSciNet: MR2100684
Digital Object Identifier: 10.2140/agt.2004.4.861

Primary: 20F67
Secondary: 20F36, 20F65, 53C23, 57M20

Rights: Copyright © 2004 Mathematical Sciences Publishers


Vol.4 • No. 2 • 2004
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