In this work, we study automorphisms of synchronous self-similar groups and the existence of extensions to continuous automorphisms over the closure of these groups with respect to the depth metric. We obtain conditions for the continuity of such extensions, but we also construct examples of groups where such extensions do not exist. We study in detail the case of the lamplighter group Lk D Zk o Z.
Matucci, F., Silva, P. (2021). Extensions of automorphisms of self-similar groups. JOURNAL OF GROUP THEORY, 1-41 [10.1515/jgth-2020-0034].
Extensions of automorphisms of self-similar groups
Matucci, F.
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2021
Abstract
In this work, we study automorphisms of synchronous self-similar groups and the existence of extensions to continuous automorphisms over the closure of these groups with respect to the depth metric. We obtain conditions for the continuity of such extensions, but we also construct examples of groups where such extensions do not exist. We study in detail the case of the lamplighter group Lk D Zk o Z.
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