Automorphisms of Higher Rank Lamplighter Groups

2014/12/06 by Melanie Stein, Stein, Melanie, Jennifer Taback +3
Mathematics · #FOS: Mathematics #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1412.2271

28 pages

arxiv created 2015/02/02 · arxiv updated 2015/02/03

Abstract

Let Γ_d(q) denote the group whose Cayley graph with respect to a particular generating set is the Diestel-Leader graph DL_d(q), as described by Bartholdi, Neuhauser and Woess. We compute both Aut(Γ_d(q)) and Out(Γ_d(q)) for d ≥ 2, and apply our results to count twisted conjugacy classes in these groups when d ≥ 3. Specifically, we show that when d ≥ 3, the groups Γ_d(q) have property R_∞, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when d=2 the lamplighter groups Γ₂(q)=L_q = \mathbb Z_q \wr \mathbb Z have property R_∞ if and only if (q,6) ≠ 1.

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