Structure of Cross-wired Lamplighter Groups

TL;DR

This paper investigates the algebraic structure of certain groups acting on Diestel-Leader graphs, proving a splitting property for specific cocompact subgroups related to lamplighter groups.

Contribution

It establishes a splitting result for cocompact subgroups of isometry groups of Diestel-Leader graphs, answering a question by Cornulier, Fisher, and Kashyap.

Findings

The sequence involving the subgroup H splits.

Identification of the unique open normal subgroup H.

Characterization of the quotient group as D_infinity.

Abstract

We answer a question posed by Cornulier, Fisher, and Kashyap, proving that for a closed cocompact subgroup $\Gamma$ of $\text{Isom}(\text{DL}(n, n))$ not contained in $\text{Isom}^+(\text{DL}(n, n))$, the sequence $1 \to H \to \Gamma \to D_\infty \to 1$ splits, where $H$ is the unique open normal subgroup such that $\Gamma/H \cong D_\infty$.