The Lamplighter Group is Not Semistable at Infinity
Michael Mihalik
TL;DR
This work resolves a long-standing question by showing the Lamplighter group $L$ is not semistable at infinity, using a van Kampen diagram analysis with $a$-bands to obtain a contradiction from carefully chosen loops and relators. It introduces a programmatic approach via $\Gamma(G,S,\mathcal{R})$ to study semistability and discusses potential generalizations through Osin’s framework of exact sequences with locally finite kernels and free quotients. It also analyzes the Extended Lamplighter Group $E$, an ascending HNN extension of $L$ that is finitely presented and semistable (indeed simply connected at infinity per Silkin), illustrating limits to constructing non-semistable examples via such extensions. Together, the results point to structural mechanisms behind semistability and suggest directions for identifying non-semistable finitely generated groups beyond $L$.
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.