Visual boundaries of Diestel-Leader graphs

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.