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

Fujiwara, Koji¹ ; Shioya, Takashi¹ ; Yamagata, Saeko¹

¹Mathematics Institute, Tohoku University, Sendai 980-8578, Japan

Algebraic and Geometric Topology, Tome 4 (2004) no. 2, pp. 861-892

Résumé

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 Hℂn. 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.

Détail

DOI : 10.2140/agt.2004.4.861

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

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