Infinite groups with fixed point properties

Abstract

We construct finitely generated groups with strong fixed point properties. Let ${\mathcal{X}}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod–$p$ acyclic for at least one prime $p$. We produce the first examples of infinite finitely generated groups $Q$ with the property that for any action of $Q$ on any $X\in {\mathcal{X}}_{ac}$, there is a global fixed point. Moreover, $Q$ may be chosen to be simple and to have Kazhdan’s property (T). We construct a finitely presented infinite group $P$ that admits no nontrivial action on any manifold in ${\mathcal{X}}_{ac}$. In building $Q$, we exhibit new families of hyperbolic groups: for each $n\ge 1$ and each prime $p$, we construct a nonelementary hyperbolic group $G_{n,p}$ which has a generating set of size $n+2$, any proper subset of which generates a finite $p$–group.