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:
[1] | DOI: 10.1016/S1570-7954(03)80078-5 · doi:10.1016/S1570-7954(03)80078-5 |
[2] | DOI: 10.1007/s00013-010-0169-2 · Zbl 1206.20034 · doi:10.1007/s00013-010-0169-2 |
[3] | Bondarenko I., Algebra Discrete Math. 1 pp 1– (2008) |
[4] | DOI: 10.1007/978-1-4612-1380-2_4 · doi:10.1007/978-1-4612-1380-2_4 |
[5] | DOI: 10.1007/3-7643-7447-0_5 · doi:10.1007/3-7643-7447-0_5 |
[6] | Grigorchuk R., CRM Proc. Lecture Notes 42 pp 225– (2007) |
[7] | DOI: 10.1017/CBO9780511626302 · Zbl 1106.37301 · doi:10.1017/CBO9780511626302 |
[8] | DOI: 10.1090/surv/117 · doi:10.1090/surv/117 |
[9] | DOI: 10.1007/s10711-006-9106-8 · Zbl 1169.20015 · doi:10.1007/s10711-006-9106-8 |
[10] | DOI: 10.1007/s13373-011-0007-2 · Zbl 1262.20032 · doi:10.1007/s13373-011-0007-2 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.