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

TL;DR

This paper fully characterizes the horofunction boundary of the lamplighter group $L_2$ with the Diestel-Leader metric, revealing a complex structure with embedded Cantor sets and non-Busemann points that are fixed under group action.

Contribution

It provides a complete description of the horofunction boundary for $L_2$ with the Diestel-Leader metric, including the structure of Busemann and non-Busemann points and their relation to the visual boundary.

Findings

The horofunction boundary $oundary_h L_2$ is fully described with a stratification.

The visual boundary $oundary_ ext{infty} L_2$ embeds into $oundary_h L_2$ as two punctured Cantor sets.

Height functions are non-Busemann horofunctions and fixed points under the group action.

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 $L_2$.