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Journal of Group Theory

Editor-in-Chief: Parker, Christopher W.

Managing Editor: Wilson, John S. / Khukhro, Evgenii I. / Kramer, Linus

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Volume 16, Issue 1


Commensurated subgroups and ends of groups

Greg Conner / Michael Mihalik
Published Online: 2013-01-03 | DOI: https://doi.org/10.1515/jgt-2012-0035


If G is a group, then subgroups A and B are commensurable if has finite index in both A and B. The commensurator of A in G, denoted by , is

It is straightforward to check that is a subgroup of G. A subgroup A is commensurated in G if . The centralizer of A in G is a subgroup of the normalizer of A in G which is a subgroup of . We develop geometric versions of commensurators in finitely generated groups. In particular, iff the Hausdorff distance between A and is finite. We show a commensurated subgroup of a group is the kernel of a certain map, and a subgroup of a finitely generated group is commensurated iff a Schreier (left) coset graph is locally finite. The ends of this coset graph correspond to the filtered ends of the pair (G,A). This last equivalence is particularly useful for deriving asymptotic results for finitely generated groups. Our primary goals in this paper are to develop and compare the basic theory of commensurated subgroups to that of normal subgroups, and to initiate the development of the asymptotic theory of commensurated subgroups.

About the article

Received: 2011-12-12

Revised: 2012-08-13

Published Online: 2013-01-03

Published in Print: 2013-01-01

Citation Information: Journal of Group Theory, Volume 16, Issue 1, Pages 107–139, ISSN (Online) 1435-4446, ISSN (Print) 1433-5883, DOI: https://doi.org/10.1515/jgt-2012-0035.

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Gregory R Conner and Michael L Mihalik
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