Vol. 285, No. 2, 2016

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Commensurations and metric properties of Houghton's groups

José Burillo, Sean Cleary, Armando Martino and Claas E. Röver

Vol. 285 (2016), No. 2, 289–301
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.

Keywords
Houghton's groups, commensurations
Mathematical Subject Classification 2010
Primary: 20F65
Milestones
Received: 22 February 2016
Accepted: 18 May 2016
Published: 21 November 2016
Authors
José Burillo
Departament de Matemàtica Aplicada IV
EETAC-UPC
Esteve Terrades 5
08860 Castelldefels
Spain
Sean Cleary
Department of Mathematics R8133
The City College of New York
160 Convent Avenue
New York, NY 10031
United States
Armando Martino
Mathematical Sciences
University of Southampton
University Road
Southampton SO17 1BJ
United Kingdom
Claas E. Röver
School of Mathematics, Statistics and Applied Mathematics
NUI Galway
University Road
Galway
Ireland