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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

We construct finitely generated groups with strong fixed point properties. Let Xac 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 Xac, 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 Xac. In building Q, we exhibit new families of hyperbolic groups: for each n 1 and each prime p, we construct a nonelementary hyperbolic group Gn,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

acyclic spaces, Kazhdan's property T, relatively hyperbolic group, simplices of groups
Mathematical Subject Classification 2000
Primary: 20F65, 20F67
Secondary: 57S30, 55M20
Received: 26 September 2008
Revised: 13 December 2008
Accepted: 12 January 2009
Published: 5 February 2009
Proposed: Walter Neumann
Seconded: Wolfgang Lueck, Benson Farb
Goulnara Arzhantseva
Université de Genève
Section de Mathématiques
2-4 rue du Lièvre
Case postale 64
1211 Genève 4
Martin R Bridson
Mathematical Institute
24-29 St Giles’
Tadeusz Januszkiewicz
Department of Mathematics
The Ohio State University
231 W 18th Ave,
Columbus, OH 43210
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
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