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Commensurating HNN extensions: nonpositive curvature and biautomaticity

Ian J Leary and Ashot Minasyan

Geometry & Topology 25 (2021) 1819–1860

We show that the commensurator of any quasiconvex abelian subgroup in a biautomatic 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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commensurating HNN extension, biautomatic group, CAT(0) group
Mathematical Subject Classification 2010
Primary: 20F10, 20F67
Secondary: 20E06
Received: 23 July 2019
Revised: 18 June 2020
Accepted: 20 June 2020
Published: 12 July 2021
Proposed: Ian Agol
Seconded: Mladen Bestvina, Benson Farb
Ian J Leary
School of Mathematical Sciences
University of Southampton
United Kingdom
Ashot Minasyan
School of Mathematical Sciences
University of Southampton
United Kingdom