Lamplighters Groups: Their Presentations, Caylely Graphs, and Generating Automata




Miller, Holly

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The focus of this paper is the traditional lamplighter group L2 = (Z/2Z) wrZ and the broader class of lamplighter groups LG = Gwr Z, where G is a finite group. These groups have seen direct use in papers such as [7], in which Grigorchuk, Linnell, Schick, and Z˙ uk show there is a closed Riemannian manifold (with the HNN-extension of L2 as its fundamental group) that has a rational L2-Betti number. Other results rely on generalizing or building atop the lamplighter group, as is done by Brieussel and Zheng in [3] to expand the known possible speed, entropy, isoperimetric profile, return probability, and compression gap attainable by random walks on finitely generated groups. This paper provides the algebraic background needed to understand the definition of the traditional lamplighter group, as well as its presentation, Caylely graph, and representation as an automaton group. All three of these constructions provide distinct avenues to generalize the lamplighter group. The sort of generalization discussed in this paper—moving from L2 to LG—is quite natural from the perspective of the lamplighter group’s Cayley graph and presentation. The case of finding a suitable automaton to generate such generalizations is more interesting; only some lamplighter groups LG can be generated by automata. When G is a finite Abelian group, then LG can be generated by the Cayley machine C(G) of G, and when G is a finite Non-Abelian group there is no automaton which can generate the group. These Cayley machines are interesting in their own right, so we explore the more general structure of the groups generated by Cayley machines.