Commensurating HNN extensions: nonpositive curvature and biautomaticity

Leary, Ian J; Minasyan, Ashot

doi:10.2140/gt.2021.25.1819

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Commensurating HNN extensions: nonpositive curvature and biautomaticity

Ian J Leary and Ashot Minasyan

Geometry & Topology 25 (2021) 1819–1860

DOI: 10.2140/gt.2021.25.1819

Abstract

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.

Keywords

commensurating HNN extension, biautomatic group, CAT(0) group

Mathematical Subject Classification 2010

Primary: 20F10, 20F67

Secondary: 20E06

References

Publication

Received: 23 July 2019

Revised: 18 June 2020

Accepted: 20 June 2020

Published: 12 July 2021

Proposed: Ian Agol

Seconded: Mladen Bestvina, Benson Farb

Authors

Ian J Leary
School of Mathematical Sciences University of Southampton Southampton United Kingdom

Ashot Minasyan
School of Mathematical Sciences University of Southampton Southampton United Kingdom

Volume 25, issue 4 (2021)