Infinite groups with fixed point properties

Journal article

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 non-trivial action by diffeomorphisms on any smooth manifold in $\mathcal{X}_{ac}$. In building $Q$, we exhibit new families of hyperbolic groups: for each $n\geq 1$ and each prime $p$, we construct a non-elementary hyperbolic group $G_{n,p}$ which has a generating set of size $n+2$, any proper subset of which generates a finite $p$-group.

Authors: Arzhantseva, G; Bridson, M; Januszkiewicz, T; Leary, I; Minasyan, A

• Publication status: Published
• Journal: Geom. Topol.
• Volume: 13
• Issue: 3
• Pages: 1229-1263
• Publication date: 2007-11-27
• DOI: 10.2140/gt.2009.13.1229
• EISSN: 1364-0380
• ISSN: 1364-0380
• Language: English
• Keywords: 20F65, 20F67; 57Sxx; math.GR; math.GT