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Branch groups. (English) Zbl 1140.20306
Hazewinkel, M. (ed.), Handbook of algebra. Volume 3. Amsterdam: Elsevier (ISBN 0-444-51264-0/hbk). 989-1112 (2003).
Summary: This is a long introduction to the theory of “branch groups”: groups acting on rooted trees which exhibit some self-similarity features in their lattice of subgroups.
Contents: 0. Introduction, 0.1. Just-infinite groups, 0.2. Algorithmic aspects, 0.3. Group presentations, 0.4. Burnside groups, 0.5. Subgroups of branch groups, 0.6. Lie algebras, 0.7. Growth, 0.8. Some notation, Acknowledgments. Part 1. Basic definitions and examples, 1. Branch groups and spherically homogeneous trees, 1.1. Algebraic definition of a branch group, 1.2. Spherically homogeneous rooted trees, 1.3. Geometric definition of a branch group, 1.4. Portraits and branch portraits, 1.5. Groups of finite automata and recursively defined automorphisms, 1.6. Examples of branch groups, 2. Spinal Groups, 2.1. Construction, basic tools and properties, 2.2. G groups, 2.3. GGS groups. Part 2. Algorithmic aspects, 3. Word and conjugacy problem, 3.1. The word problem, 3.2. The conjugacy problem in \(\mathfrak G\). 4. Presentations and endomorphic presentations of branch groups. 4.1. Nonfinite presentability, 4.2. Endomorphic presentations of branch groups, 4.3. Examples. Part 3. Algebraic aspects, 5. Just-infinite branch groups, 6. Torsion branch groups, 7. Subgroup structure, 7.1. The derived series, 7.2. The powers series, 7.3. Parabolic subgroups, 7.4. The structure of \(\mathfrak G\), 7.5. The structure of \(\Gamma\), 7.6. The structure of \(\overline\Gamma\), 7.7. The structure of \(\overline{\overline\Gamma}\). 8. Central series, finiteness of width and associated Lie algebras, 8.1. \(N\)-series, 8.2. Lie algebras of branch groups, 8.3. Subgroup growth. 9. Representation theory of branch group. Part 4. Geometric and analytic aspects, 10. Growth, 10.1. Growth of G groups with finite directed part, 10.2. Growth of G groups defined by homogeneous sequences, 10.3. Parabolic space and Schreier graphs, 11. Spectral properties of unitary representations, 11.1. Unitary representations, 11.2. Operator recursions, 12. Open problems. References.
For the entire collection see [Zbl 1052.00009].

MSC:
20E08 Groups acting on trees
20F50 Periodic groups; locally finite groups
20F65 Geometric group theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
22D10 Unitary representations of locally compact groups
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
43A07 Means on groups, semigroups, etc.; amenable groups
68Q70 Algebraic theory of languages and automata
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