Bounded Depth Ascending HNN Extensions and $\unicode[STIX]{x1D70B}_{1}$-Semistability at infinity

Mihalik, Michael L.

Canadian journal of mathematics, Tome 72 (2020) no. 6, pp. 1529-1550

Résumé

A well-known conjecture is that all finitely presented groups have semistable fundamental groups at infinity. A class of groups whose members have not been shown to be semistable at infinity is the class ${\mathcal{A}}$ of finitely presented groups that are ascending HNN-extensions with finitely generated base. The class ${\mathcal{A}}$ naturally partitions into two non-empty subclasses, those that have “bounded” and “unbounded” depth. Using new methods introduced in a companion paper we show those of bounded depth have semistable fundamental group at infinity. Ascending HNN extensions produced by Ol’shanskii–Sapir and Grigorchuk (for other reasons), and once considered potential non-semistable examples are shown to have bounded depth. Finally, we devise a technique for producing explicit examples with unbounded depth. These examples are perhaps the best candidates to date in the search for a group with non-semistable fundamental group at infinity.

Détail

DOI : 10.4153/S0008414X19000385

Mots-clés : ascending HNN extension, fundamental group at infinity, semistability at infinity