Commensurating HNN extensions: nonpositive 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 $CAT\left(0\right)$ but not biautomatic. These groups also resolve a number of other questions concerning $CAT\left(0\right)$ groups.