Splittings and the asymptotic topology of the lamplighter group

TL;DR

This paper investigates the geometric properties of the lamplighter group, revealing that its Cayley graph is coarsely separated by quasi-circles, contrasting with known characterizations for finitely presented groups.

Contribution

It demonstrates that the geometric characterization of splittings over 2-ended groups does not extend to all finitely generated groups, specifically analyzing the lamplighter group.

Findings

Cayley graph of lamplighter group is coarsely separated by quasi-circles

Characterization of group splittings over 2-ended groups does not generalize to all finitely generated groups

Provides an answer to Kleiner's question about the lamplighter group

Abstract

It is known that splittings of finitely presented groups over 2-ended groups can be characterized geometrically. We show that this characterization does not extend to all finitely generated groups. Answering a question of Kleiner we show that the Cayley graph of the lamplighter group is coarsely separated by quasi-circles.