Infinite groups with fixed point properties

Infinite groups with fixed point properties

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.

acyclic spaces, kazhdan's property t, relatively hyperbolic group, simplices of groups

1229-1263

Arzhantseva, Goulnara

cb099fac-8639-46ef-bac8-a93f68309758

Bridson, Martin R.

28b22b3b-2ff1-4b35-a3b7-cc2fd559e689

Januszkiewicz, Tadeusz

a29c937f-86d4-4d11-adf0-891e10933a0a

Leary, Ian J.

57bd5c53-cd99-41f9-b02a-4a512d45150e

Minasyan, Ashot

3de640f5-d07b-461f-b130-5b1270bfdb3d

Swiatkowski, Jacek

95730838-e5a6-4e3e-bbb3-a5d04ee02b7f

5 February 2009

Arzhantseva, Goulnara

cb099fac-8639-46ef-bac8-a93f68309758

Bridson, Martin R.

28b22b3b-2ff1-4b35-a3b7-cc2fd559e689

Januszkiewicz, Tadeusz

a29c937f-86d4-4d11-adf0-891e10933a0a

Leary, Ian J.

57bd5c53-cd99-41f9-b02a-4a512d45150e

Minasyan, Ashot

3de640f5-d07b-461f-b130-5b1270bfdb3d

Swiatkowski, Jacek

95730838-e5a6-4e3e-bbb3-a5d04ee02b7f

Arzhantseva, Goulnara, Bridson, Martin R., Januszkiewicz, Tadeusz, Leary, Ian J., Minasyan, Ashot and Swiatkowski, Jacek
(2009)
Infinite groups with fixed point properties.
*Geometry & Topology*, 13 (3), .
(doi:10.2140/gt.2009.13.1229).

## Abstract

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.

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## More information

Published date: 5 February 2009

Keywords:
acyclic spaces, kazhdan's property t, relatively hyperbolic group, simplices of groups

Organisations:
Pure Mathematics

## Identifiers

Local EPrints ID: 349646

URI: http://eprints.soton.ac.uk/id/eprint/349646

ISSN: 1465-3060

PURE UUID: 1cd93f42-95ef-4a50-bafe-053d79012bc0

## Catalogue record

Date deposited: 11 Mar 2013 10:12

Last modified: 17 Dec 2019 01:43

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## Contributors

Author:
Goulnara Arzhantseva

Author:
Martin R. Bridson

Author:
Tadeusz Januszkiewicz

Author:
Jacek Swiatkowski

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