Structure of Cross-wired Lamplighter Groups

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 $\Gamma /H\cong D_{\infty }$.