Commensurating HNN extensions: nonpositive curvature and biautomaticity

Leary, Ian J. and Minasyan, Ashot (2021) Commensurating HNN extensions: nonpositive curvature and biautomaticity. Geometry and Topology, 25 (4), 1819-1860. (doi:10.2140/gt.2021.25.1819).

Record type: Article

Abstract

We show that the commensurator of any quasiconvex abelian subgroup in a biauto-matic group is small, in the sense that it has finite image in the abstract commensurator of the subgroup. Using this criterion we exhibit groups that are CAT.0/ but not biautomatic. These groups also resolve a number of other questions concerning CAT.0/ groups.

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Accepted/In Press date: 18 June 2020

e-pub ahead of print date: 12 July 2021

Additional Information: Funding Information: Acknowledgements First, the authors thank Tomasz Prytuła. He asked the authors for an example of an abelian subgroup of a CAT.0/ group whose commensurator does not normalize any finite-index subgroup, in connection with his work on classifying spaces for families of abelian subgroups [33; 35]. This question was what originally led us to consider the groups GP0 and GP. The authors also thank Martin Bridson, Pierre-Emmanuel Caprace, Derek Holt, Jingyin Huang, Denis Osin and Kevin Whyte for helpful comments on aspects of the work. Leary was partially supported by a research fellowship from the Leverhulme Trust. Most of the work was done in Southampton. However, during the project, Leary spent some weeks at INI, Cambridge, where research was supported by EPSRC grant EP/K032208/1.

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