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Commensurations and metric properties of Houghton's groups

Commensurations and metric properties of Houghton's groups
Commensurations and metric properties of Houghton's groups
We describe the automorphism groups and the abstract commensurators of Houghton’s groups. Then we give sharp estimates for the word metric of these groups and deduce that the commensurators embed into the corresponding quasi-isometry groups. As a further consequence, we obtain that the Houghton group on two rays is at least quadratically distorted in those with three or more rays.
Houghton's groups, commensurations
0030-8730
289-301
Burillo, Jose
e758157e-f040-402e-a011-2cb317f85771
Cleary, Sean
dfc10845-e8ce-4d6b-a31a-0a24a4492ba4
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Rover, Claas
491d8bbd-be40-4ce7-9f6d-82ba483f46da
Burillo, Jose
e758157e-f040-402e-a011-2cb317f85771
Cleary, Sean
dfc10845-e8ce-4d6b-a31a-0a24a4492ba4
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Rover, Claas
491d8bbd-be40-4ce7-9f6d-82ba483f46da

Burillo, Jose, Cleary, Sean, Martino, Armando and Rover, Claas (2016) Commensurations and metric properties of Houghton's groups. Pacific Journal of Mathematics, 285 (2), 289-301. (doi:10.2140/pjm.2016.285.289).

Record type: Article

Abstract

We describe the automorphism groups and the abstract commensurators of Houghton’s groups. Then we give sharp estimates for the word metric of these groups and deduce that the commensurators embed into the corresponding quasi-isometry groups. As a further consequence, we obtain that the Houghton group on two rays is at least quadratically distorted in those with three or more rays.

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Accepted/In Press date: 18 May 2016
e-pub ahead of print date: 21 November 2016
Published date: 21 November 2016
Keywords: Houghton's groups, commensurations
Organisations: Pure Mathematics

Identifiers

Local EPrints ID: 405611
URI: https://eprints.soton.ac.uk/id/eprint/405611
ISSN: 0030-8730
PURE UUID: e0737f0a-3108-4432-a1e9-06f7137fac18

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Date deposited: 08 Feb 2017 15:39
Last modified: 09 Jan 2018 17:33

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Contributors

Author: Jose Burillo
Author: Sean Cleary
Author: Armando Martino
Author: Claas Rover

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