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.
"Parabolic isometries of CAT(0) spaces and CAT(0) dimensions." Algebr. Geom. Topol. 4 (2) 861 - 892, 2004. https://doi.org/10.2140/agt.2004.4.861