The horofunction boundary of the lamplighter group $L_2$ with the Diestel-Leader metric

Abstract

We fully describe the horofunction boundary ${\partial }_h{L}_2$ with the word metric associated with the generating set $\{t,at\}$ (i.e the metric arising in the Diestel-Leader graph $\text{DL}(2,2)$). The visual boundary ${\partial }_{\infty }{L}_2$ with this metric is a subset of ${\partial }_h{L}_2$. Although ${\partial }_{\infty }{L}_2$ does not embed continuously in ${\partial }_h{L}_2$, it naturally splits into two subspaces, each of which is a punctured Cantor set and does embed continuously. The height function on $\text{DL}(2,2)$ provides a natural stratification of ${\partial }_h{L}_2$, in which countably-many non-Busemann points interpolate between the two halves of ${\partial }_{\infty }{L}_2$. Furthermore, the height function and its negation are themselves non-Busemann horofunctions in ${\partial }_h{L}_2$ and are global fixed points of the action of $L_2$.

18 pages, 4 figures