Commensurating HNN-extensions: non-positive curvature and biautomaticity

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.

v4: 41 pages, re-formatted in G&T style. Some misprints have been corrected and references to recent results by other authors have been added

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• Higher-rank GBS groups: non-positive curvature and biautomaticity
• Isomorphism classification of Leary-Minasyan groups