# 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.

## Key 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.