#### Volume 25, issue 4 (2021)

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Commensurating HNN extensions: nonpositive curvature and biautomaticity

### Ian J Leary and Ashot Minasyan

Geometry & Topology 25 (2021) 1819–1860
##### 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 $CAT\left(0\right)$ but not biautomatic. These groups also resolve a number of other questions concerning $CAT\left(0\right)$ groups.

##### Keywords
commensurating HNN extension, biautomatic group, CAT(0) group
##### Mathematical Subject Classification 2010
Primary: 20F10, 20F67
Secondary: 20E06