On finite generation of self-similar groups of finite type. (English) Zbl 1280.20030
Summary: A self-similar group of finite type is the profinite group of all automorphisms of a regular rooted tree that locally around every vertex act as elements of a given finite group of allowed actions. We provide criteria for determining when a self-similar group of finite type is finite, level-transitive, or topologically finitely generated. Using these criteria and GAP computations we show that for the binary alphabet there is no infinite topologically finitely generated self-similar group given by patterns of depth 3, and there are 32 such groups for depth 4.
MSC:
| 20E08 | Groups acting on trees |
| 20E18 | Limits, profinite groups |
| 20F05 | Generators, relations, and presentations of groups |
| 20F65 | Geometric group theory |
Keywords:
self-similar groups; topologically finitely generated groups; branch groups; profinite groups; automorphisms of regular rooted treesSoftware:
GAPReferences:
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