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From fractal groups to fractal sets. (English) Zbl 1037.20040
Grabner, Peter (ed.) et al., Fractals in Graz 2001. Analysis, dynamics, geometry, stochastics. Proceedings of the conference, Graz, Austria, June 2001. Basel: Birkhäuser (ISBN 3-7643-7006-8/hbk). Trends in Mathematics, 25-118 (2003).
The authors present a survey of ideas, notions and results that are connected to ‘self-similarity’ of groups, semigroups and their actions. They also exhibit new connections of groups and semigroups with fractal objects, in particular with Julia sets.
Originally, Grigorchuk constructed a finitely generated group \(G\) of automorphisms of the segment \([0,1]\) without dyadically rational points. The generators were defined as permutations of subintervals. One of the main properties of this action is that if we restrict to a subinterval \(I=[(k-1)/2^n,k/2^n]\) the action of its stabilizer \(G_I=\roman{st}_G(I)\), then the restriction will coincide with the action of \(G\) on \([0,1]\) (after natural identification). Another fundamental property is the fact that the action of the group \(G\) is ‘contracting’, i.e. the canonical homomorphisms \(G_I\to G\) contract the length of the group elements by a constant \(\lambda>1\). The third fundamental property is the ‘branching’ nature of the action. This means that, up to finite-index inclusions, the stabilizers of the partition into the dyadic subintervals of the \(n\)-th level are direct products of \(2^n\) isomorphic groups and that the lattice of subnormal subgroups has a tree-like structure. Later, other similar examples of groups were constructed. It became clear that they belong to the big class of ‘finitely automatic groups’. Recently, Grigorchuk introduced the class of ‘branch groups’. A regular rooted tree is an example of a geometric object most closely related with the notion of self-similarity. The following properties are exposed in detail.
(i) The authors discuss the notion of a self-similar set and define the notion of a self-similar action of a group and of an inverse semigroup. One main source of examples of self-similar group actions are the actions generated by finite automata.
(ii) Strong contraction property and branch property of the Grigorchuk group imply the intermediate growth of this group. Such problems can also be related with problems of fractal geometry, in particular with the problems of computation of the Hausdorff dimension. This leads to a study of formal languages over a finite alphabet. The authors discuss the growth of formal languages, groups, semigroups, graphs, and finite automata. Self-similar groups also appear in the study of the Hausdorff dimension of profinite groups.
(iii) Schreier graphs of self-similar groups have very interesting spectral properties. Their limit spaces often have a fractal nature. Recently, Nekrashevich introduced the notion of an iterated monodromy group. The Schreier graphs of contacting self-similar groups converge to nice fractal topological spaces. This is formalized by the notion of the limit space of a self-similar action.
(iv) An important role in the study of self-similar groups is played by ‘virtual endomorphisms’, i.e. endomorphisms defined on a subgroup of finite index. Lysenok obtained a finite \(L\)-presentation of the Grigorchuk group, i.e. a presentation in which the defining relations are obtained from a finite set of relations using iterated application of a substitution \(f\) over the alphabet of generators. Branch groups have such \(L\)-presentations.
(v) ‘Boundaries’. One of the sources of fractal sets are the various boundaries of groups. The authors present a short survey of notions and facts about hyperbolic spaces, groups, and their boundaries. The notion of self-similarity (of topological spaces) and the notion of a self-similar action is interpreted from a common point of view using the notion of a Hilbert bimodule over a \(C^*\)-algebra.
For the entire collection see [Zbl 1005.00032].

MSC:
20F65 Geometric group theory
20E18 Limits, profinite groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20E08 Groups acting on trees
20E07 Subgroup theorems; subgroup growth
28A80 Fractals
37F50 Small divisors, rotation domains and linearization in holomorphic dynamics
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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