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Infinite groups with fixed point properties

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
1465-3060
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
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), 1229-1263. (doi:10.2140/gt.2009.13.1229).

Record type: Article

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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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: https://eprints.soton.ac.uk/id/eprint/349646
ISSN: 1465-3060
PURE UUID: 1cd93f42-95ef-4a50-bafe-053d79012bc0
ORCID for Ian J. Leary: ORCID iD orcid.org/0000-0001-8300-4979
ORCID for Ashot Minasyan: ORCID iD orcid.org/0000-0002-4986-2352

Catalogue record

Date deposited: 11 Mar 2013 10:12
Last modified: 24 Sep 2019 00:43

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