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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


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 2. 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 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.

CAT(0) space, parabolic isometry, Artin group, Heisenberg group, geometric dimension, cohomological dimension
Mathematical Subject Classification 2000
Primary: 20F67
Secondary: 20F65, 20F36, 57M20, 53C23
Forward citations
Received: 17 September 2003
Revised: 30 July 2004
Accepted: 13 September 2004
Published: 9 October 2004
Koji Fujiwara
Mathematics Institute
Tohoku University
Sendai 980-8578
Takashi Shioya
Mathematics Institute
Tohoku University
Sendai 980-8578
Saeko Yamagata
Mathematics Institute
Tohoku University
Sendai 980-8578