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(0) but not biautomatic. These groups also resolve a number of other questions concerning CAT(0) groups.

A survey on quasi-isometries of pairs: invariants and rigidity

“…Note that the generalisation to Z n -by-(8 ended) group fails if n ě 2. Indeed, Leary-Minasyan have constructed groups quasi-isometric to Z n ˆF2 which do not have a normal Z n -subgroup [LM21]. More examples quasi-isometric to certain RAAGs or right-angled buildings were constructed by the first author in [Hug21] (see also [Hug22]).…”

Sausages

“…One piece of evidence in favor of this is that S 3 (and, for trivial reasons, S 2 ) is homotopy equivalent to a locally CAT(0) complex, and it is plausible that the same holds for all S q . Although there are known examples of groups which are locally CAT(0) but not bi-automatic [20], nevertheless in practice these two properties often go hand in hand.…”

Commensurating HNN-extensions: hierarchical hyperbolicity and biautomaticity

“…In the 1990s Alonso and Bridson introduced the class of semihyperbolic groups [AB95] which contains all CATp0q and biautomatic groups. In recent work of Leary and Minasyan [LM21], the authors construct irreducible uniform lattices in IsompE 2n q ˆT2m (m ě 2, n ě 1), giving the first examples of CATp0q groups which are not biautomatic. These groups were classified up to isomorphism by the second author [Val21a] and studied in the context of fibring by the first author [Hug22].…”

“…The group we construct is a "hyperbolic" analogue of the groups introduced by Leary-Minasyan in [LM21]. Indeed, Γ is an HNN-extension of an arithmetic surface where the stable letter commensurates the surface whilst acting as an infinite order elliptic isometry of the hyperbolic plane RH 2 .…”