Geometry & Topology

Infinite groups with fixed point properties

Goulnara Arzhantseva, Martin R Bridson, Tadeusz Januszkiewicz, Ian J Leary, Ashot Minasyan, and Jacek None

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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 XXac, 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 n1 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.

Article information

Geom. Topol., Volume 13, Number 3 (2009), 1229-1263.

Received: 26 September 2008
Revised: 13 December 2008
Accepted: 12 January 2009
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F67: Hyperbolic groups and nonpositively curved groups
Secondary: 57S30: Discontinuous groups of transformations 55M20: Fixed points and coincidences [See also 54H25]

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


Arzhantseva, Goulnara; Bridson, Martin R; Januszkiewicz, Tadeusz; Leary, Ian J; Minasyan, Ashot; None, Jacek. Infinite groups with fixed point properties. Geom. Topol. 13 (2009), no. 3, 1229--1263. doi:10.2140/gt.2009.13.1229.

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