{"paper":{"title":"Automorphisms of Higher Rank Lamplighter Groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Jennifer Taback, Melanie Stein, Peter Wong","submitted_at":"2014-12-06T19:59:46Z","abstract_excerpt":"Let $\\Gamma_d(q)$ 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 $Aut(\\Gamma_d(q))$ and $Out(\\Gamma_d(q))$ for $d \\geq 2$, and apply our results to count twisted conjugacy classes in these groups when $d \\geq 3$. Specifically, we show that when $d \\geq 3$, the groups $\\Gamma_d(q)$ have property $R_{\\infty}$, that is, every automorphism has an infinite number of twisted conjugacy classes. In contrast, when $d=2$ the lamplighter groups $\\Gamma_2(q)=L_q = {\\mathb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.2271","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}