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

Abstract

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 $CAT\left(0\right)$ spaces of dimension $2$ by isometries, although such actions exist on $CAT\left(0\right)$ 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 $n\ge 2$. The group $G$ is acting on the universal cover of $M$, which is isometric to $H_{ℂ}^n$. It is a $CAT\left(-1\right)$ space of dimension $2n$. The geometric dimension of $G$ is $2n-1$. We show that $G$ does not act on any proper $CAT\left(0\right)$ 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.