Torsion subgroups and fixed-point rigidity in CAT(0) geometry

Abstract

We develop new methods for studying groups acting on CAT(0) spaces, which lead to several general structural results. First, we prove that every torsion subgroup of a CAT(0) group is finite, resolving a question of Swenson from the 1990s. The proof is based on showing that random walks on any finitely generated torsion group with bounded exponent acting on a CAT(0) space have zero drift. This is then combined with the fixed-point rigidity that we develop. Second, we show that any finitely generated torsion group of bounded exponent has a global fixed point whenever it acts properly by isometries on a CAT(0) space of bounded geometry, or, without the properness assumption, by isometries on a finite-dimensional CAT(0) space. Third, we establish a Kazhdan-type rigidity principle that underlies many of our results: let $\Gamma$ be a finitely generated group such that every isometric action of $\Gamma$ on $\mathbb{R}^n$ has a fixed point. Then every fixed-point-free action of $\Gamma$ on a geodesically complete $n$-dimensional CAT(0) space of bounded geometry has joint minimal displacement uniformly bounded away from zero. In particular, almost fixed points imply a global fixed point. This applies in particular to groups with property (T), torsion groups, certain branch groups, and mapping class groups. Fourth, we establish the following alternative for any finitely generated amenable group: either every action on a finite-dimensional CAT(0) space has a global fixed point, or the group has non-vanishing virtual first Betti number. Further consequences include that finitely generated torsion groups cannot act without a global fixed point on geodesically complete CAT(0) spaces of bounded geometry that are either visibility spaces or have compact Tits boundary. The methods involve scalings of actions by ultralimits and random walks.