# Commensurating HNN-extensions: non-positive curvature and biautomaticity

## TL;DR

This paper investigates the properties of certain groups, showing that the commensurator of quasiconvex abelian subgroups in biautomatic groups is limited, and constructs examples of CAT(0) groups that are not biautomatic, addressing key questions in geometric group theory.

## Contribution

It establishes a new criterion relating commensurators to biautomaticity and provides explicit examples of CAT(0) groups lacking biautomaticity.

## Key findings

- The commensurator of quasiconvex abelian subgroups in biautomatic groups has finite image.
- Existence of CAT(0) groups that are not biautomatic.
- Resolution of open questions about the relationship between CAT(0) and biautomatic groups.

## 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.