#### Volume 13, issue 3 (2009)

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Infinite groups with fixed point properties

### Goulnara Arzhantseva, Martin R Bridson, Tadeusz Januszkiewicz, Ian J Leary, Ashot Minasyan and Jacek Światkowski

Geometry & Topology 13 (2009) 1229–1263
##### Abstract

We construct finitely generated groups with strong fixed point properties. Let ${\mathsc{X}}_{ac}$ 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 $X\in {\mathsc{X}}_{ac}$, 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 ${\mathsc{X}}_{ac}$. 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.

 Dedicated to Michael W Davis on the occasion of his 60th birthday
##### Keywords
acyclic spaces, Kazhdan's property T, relatively hyperbolic group, simplices of groups
##### Mathematical Subject Classification 2000
Primary: 20F65, 20F67
Secondary: 57S30, 55M20