Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 4, Number 2 (2004), 861-892.
Parabolic isometries of CAT(0) spaces and CAT(0) dimensions
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 which do not act properly on any proper spaces of dimension by isometries, although such actions exist on spaces of dimension .
Another example is the fundamental group, , of a complete, non-compact, complex hyperbolic manifold with finite volume, of complex dimension . The group is acting on the universal cover of , which is isometric to . It is a space of dimension . The geometric dimension of is . We show that does not act on any proper space of dimension properly by isometries.
We also discuss the fundamental groups of a torus bundle over a circle, and solvable Baumslag–Solitar groups.
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
Permanent link to this document
Digital Object Identifier
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
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. https://projecteuclid.org/euclid.agt/1513882535