Rufus A. Lyman: Pharmacy's Lamplighter

Cross-wired lamplighter groups

We give a necessary and sufficient condition for a locally compact group to be isomorphic to a closed cocompact subgroup in the isometry group of a Diestel–Leader graph. As a consequence of this condition, we see that every cocompact lattice in the isometry group of a Diestel–Leader graph admits a transitive, proper action on some other Diestel–Leader graph. We also give some examples of lattic...

A Tight Bound for the Lamplighter Problem

We settle an open problem, raised by Y. Peres and D. Revelle, concerning the L mixing time of the random walk on the lamplighter graph. We also provide general bounds relating the entropy decay of a Markov chain to the separation distance of the chain, and show that the lamplighter graphs once again provide examples of tightness of our results.

Rufus Cole, 1913

Acceleration of lamplighter random walks

Suppose we are given an infinite, finitely generated group G and a transient random walk with bounded range on the wreath product (Z/2Z) ≀ G, such that its projection on G is transient. This random walk can be interpreted as a lamplighter random walk, where there is a lamp at each element of G, which can be switched on and off, and a lamplighter walks along G and switches lamps randomly on and ...

The Sigma Invariants of the Lamplighter Groups

We compute the Bieri-Neumann-Strebel-Renz geometric invariants, Σn, of the lamplighter groups Lm by using the Diestel-Leader graph DL(m,m) to represent the Cayley graph of Lm.