Bounded Depth Ascending HNN Extensions and $\pi_1$-Semistability at $\infty$

TL;DR

This paper investigates the semistability at infinity of finitely presented groups, especially focusing on ascending HNN extensions with bounded and unbounded depth, establishing semistability for the bounded case.

Contribution

It proves that bounded depth finitely presented ascending HNN extensions have semistable fundamental groups at infinity, and introduces a method to construct extensions with unbounded depth.

Findings

Bounded depth ascending HNN extensions are semistable at infinity.

A technique for constructing unbounded depth ascending HNN extensions.

Semistability holds under certain asymptotic conditions.

Abstract

A 1-ended finitely presented group has semistable fundamental group at $\infty$ if it acts geometrically on some (equivalently any) simply connected and locally finite complex $X$ with the property that any two proper rays in $X$ are properly homotopic. If $G$ has semistable fundamental group at $\infty$ then one can unambiguously define the fundamental group at $\infty$ for $G$. The problem, asking if all finitely presented groups have semistable fundamental group at $\infty$ has been studied for over 40 years. If $G$ is an ascending HNN extension of a finitely presented group then indeed, $G$ has semistable fundamental group at $\infty$, but since the early 1980's it has been suggested that the finitely presented groups that are ascending HNN extensions of {\it finitely generated} groups may include a group with non-semistable fundamental group at $\infty$. Ascending HNN extensions naturally break into two classes, those with bounded depth and those with unbounded depth. Our main theorem shows that bounded depth finitely presented ascending HNN extensions of finitely generated groups have semistable fundamental group at $\infty$. Semistability is equivalent to two weaker asymptotic conditions on the group holding simultaneously. We show one of these conditions holds for all ascending HNN extensions, regardless of depth. We give a technique for constructing ascending HNN extensions with unbounded depth. This work focuses attention on a class of groups that may contain a group with non-semistable fundamental group at $\infty$.