Infinite groups with fixed point properties

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

doi:10.2140/gt.2009.13.1229

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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

DOI: 10.2140/gt.2009.13.1229

Abstract

We construct finitely generated groups with strong fixed point properties. Let $X_{ac}$ 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 $X \in X_{ac}$ , 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 $X_{ac}$ . In building $Q$ , we exhibit new families of hyperbolic groups: for each $n \geq 1$ 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

References

Publication

Received: 26 September 2008

Revised: 13 December 2008

Accepted: 12 January 2009

Published: 5 February 2009

Proposed: Walter Neumann

Seconded: Wolfgang Lueck, Benson Farb

Authors

Goulnara Arzhantseva
Université de Genève Section de Mathématiques 2-4 rue du Lièvre Case postale 64 1211 Genève 4 Switzerland

Martin R Bridson
Mathematical Institute 24-29 St Giles’ Oxford UK

Tadeusz Januszkiewicz
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

Ian J Leary
Department of Mathematics The Ohio State University 231 W 18th Ave Columbus, OH 43210 USA

Ashot Minasyan
School of Mathematics University of Southampton Highfield, Southampton, SO17 1BJ United Kingdom

Jacek Światkowski
Instytut Matematyczny Uniwersytet Wroclawski pl. Grunwaldzki 2/4 50-384 Wroclaw Poland

Volume 13, issue 3 (2009)