Near Ascending HNN-Extensions and a Combination Result for Semistability at Infinity

TL;DR

This paper proves a new combination theorem ensuring that certain groups constructed from graphs of groups are semistable at infinity, advancing understanding of the asymptotic properties of finitely presented groups.

Contribution

It establishes a novel combination result for semistability at infinity in groups formed via graphs of groups and HNN extensions.

Findings

Groups from connected reduced graphs of groups are semistable at infinity under specified conditions.

HNN extensions of finitely presented groups with finite index subgroups are semistable at infinity.

If the base group in an HNN extension is 1-ended, the extension is simply connected at infinity.

Abstract

Semistability at infinity is an asymptotic property of finitely presented groups that is needed in order to effectively define the fundamental group at infinity for a 1-ended group. It is an open problem whether or not all finitely presented groups have semistable fundamental group at infinity. While many classes of groups are known to contain only semistable at infinity groups, there are only a few combination results for such groups. Our main theorem is such a result. Main Theorem. Suppose $G$ is the fundamental group of a connected reduced graph of groups, where each edge group is infinite and finitely generated, and each vertex group is finitely presented and either 1-ended and semistable at infinity or has an edge group of finite index. Then $G$ is 1-ended and semistable at infinity. An important part of the proof of this result is the semistability part of the following: Theorem. Suppose $H_0$ is an infinite finitely presented group, $H_1$ is a subgroup of finite index in $H_0$, $\phi:H_1\to H_0$ is a monomorphism and $G=H_0\ast_\phi$ is the resulting HNN extension. Then $G$ is 1-ended and semistable at infinity. If additionally, $H_0$ is 1-ended, then $G$ is simply connected at infinity.