The Lamplighter Group is Not Semistable at Infinity

TL;DR

This paper proves that the Lamplighter group is not semistable at infinity, providing a key example in the study of geometric group theory and the properties of finitely generated groups.

Contribution

It establishes the first confirmed example of a finitely generated group that is not semistable at infinity, advancing understanding of group boundaries.

Findings

Lamplighter group is not semistable at infinity

Finitely generated non-semistable groups can inform the construction of non-semistable finitely presented groups

Extended Lamplighter group is simply connected at infinity

Abstract

The question of whether or not all finitely presented groups are semistable at infinity has been studied for over 40 years. In 1986, we defined what it means for a finitely generated group to be semistable at infinity - in analogy with the definition for finitely presented groups. At that time we suggest that the Lamplighter group may not be semistable at infinity, but until now there was no confirmed example of a finitely generated group that is not semistable at infinity. We prove the Lamplighter group is not semistable at infinity. Finitely generated non-semistable groups may be important in finding non-semistable finitely presented groups via ascending HNN extensions. There is an ascending HNN extension E of the Lamplighter group (called the Extended Lamplighter group) that is finitely presented. It would seem that E is a candidate to be a finitely presented non-semistable at infinity group, but a result of N. Silkin, shows that E is in fact simply connected at infinity.