Algebraic & Geometric Topology

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

Koji Fujiwara, Takashi Shioya, and Saeko Yamagata

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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.

Article information

Algebr. Geom. Topol., Volume 4, Number 2 (2004), 861-892.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F36: Braid groups; Artin groups 57M20: Two-dimensional complexes 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

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


Fujiwara, Koji; Shioya, Takashi; Yamagata, Saeko. Parabolic isometries of CAT(0) spaces and CAT(0) dimensions. Algebr. Geom. Topol. 4 (2004), no. 2, 861--892. doi:10.2140/agt.2004.4.861.

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