Infinite groups with fixed point properties

Arzhantseva, Goulnara¹ ; Bridson, Martin R² ; Januszkiewicz, Tadeusz³ ; Leary, Ian J⁴ ; Minasyan, Ashot⁵ ; Światkowski, Jacek⁶

¹ Université de Genève, Section de Mathématiques, 2-4 rue du Lièvre, Case postale 64, 1211 Genève 4, Switzerland
² Mathematical Institute, 24-29 St Giles’, Oxford, UK
³ Department of Mathematics, The Ohio State University, 231 W 18th Ave,, Columbus, OH 43210, USA, and The Mathematical Institute of Polish Academy of Sciences, On leave from Instytut Matematyczny, Uniwersytet Wrocławski
⁴ Department of Mathematics, The Ohio State University, 231 W 18th Ave, Columbus, OH 43210, USA
⁵ School of Mathematics, University of Southampton, Highfield, Southampton, SO17 1BJ, United Kingdom
⁶ Instytut Matematyczny, Uniwersytet Wroclawski, pl. Grunwaldzki 2/4, 50-384 Wroclaw, Poland

Geometry & topology, Tome 13 (2009) no. 3, pp. 1229-1263.

Résumé

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.

Détail

DOI : 10.2140/gt.2009.13.1229

Keywords: acyclic spaces, Kazhdan's property T, relatively hyperbolic group, simplices of groups

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