Visual boundaries of Diestel-Leader graphs

Author name not available (Why is that?)

Publication date: 17 April 2015

Abstract: Diestel-Leader graphs are neither hyperbolic nor CAT(0), so their visual boundaries may be pathological. Indeed, we show that for

d > 2

,

p a r t i a l e x t {D L}_{d} (q)

carries the indiscrete topology. On the other hand,

p a r t i a l e x t {D L}_{2} (q)

, while not Hausdorff, is

T_{1}

, totally disconnected, and compact. Since

e x t {D L}_{2} (q)

is a Cayley graph of the lamplighter group

L_{q}

, we also obtain a nice description of

p a r t i a l e x t {D L}_{2} (q)

in terms of the lamp stand model of

L_{q}

and discuss the dynamics of the action.

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