Visual boundaries of Diestel-Leader graphs

TL;DR

This paper investigates the visual boundaries of Diestel-Leader graphs, revealing their topological properties vary with dimension, and provides a detailed description for the 2-dimensional case related to lamplighter groups.

Contribution

It characterizes the topological structure of the visual boundary of Diestel-Leader graphs, especially for the 2-dimensional case linked to lamplighter groups, highlighting differences from hyperbolic and CAT(0) spaces.

Findings

For d>2, the boundary has the indiscrete topology.

For d=2, the boundary is T1, totally disconnected, and compact.

The boundary of DL_2(q) is described via the lamp stand model of the lamplighter group.

Abstract

Diestel-Leader graphs are neither hyperbolic nor CAT(0), so their visual boundaries may be pathological. Indeed, we show that for $d>2$, $\partial\text{DL}_d(q)$ carries the indiscrete topology. On the other hand, $\partial\text{DL}_2(q)$, while not Hausdorff, is $T_1$, totally disconnected, and compact. Since $\text{DL}_2(q)$ is a Cayley graph of the lamplighter group $L_q$, we also obtain a nice description of $\partial\text{DL}_2(q)$ in terms of the lamp stand model of $L_q$ and discuss the dynamics of the action.