Volodymyr Nekrashevych

Self-similar groups

Basic definitions and examples Algebraic theory Limit spaces Orbispaces Iterated monodromy groups Examples and applications Bibliography Index.

From Fractal Groups to Fractal Sets

The idea of self-similarity is one of the most fundamental in the modern mathematics. The notion of “renormalization group”, which plays an essential role in quantum field theory, statistical physics and dynamical systems, is related to it. The notions of fractal and multi-fractal, playing an important role in singular geometry, measure theory and holomorphic dynamics, are also related. Self-similarity also appears in the theory of C*-algebras (for example in the representation theory of the Cuntz algebras) and in many other branches of mathematics. Starting from 1980 the idea of self-similarity entered algebra and began to exert great influence on asymptotic and geometric group theory.

Thurston equivalence of topological polynomials

We answer Hubbards question on determining the Thurston equivalence class of “twisted rabbits”, i.e. composita of the “rabbit” polynomial with nth powers of the Dehn twists about its ears. The answer is expressed in terms of the 4-adic expansion of n. We also answer the equivalent question for the other two families of degree-2 topological polynomials with three post-critical points. In the process, we rephrase the questions in group-theoretical language, in terms of wreath recursions.

Groups St Andrews 2009 in Bath: Iterated monodromy groups

We associate a group IMG(f) to every covering f of a topological space M by its open subset. It is the quotient of the fundamental group �1(M) by the intersection of the kernels of its monodromy action for the iterates f n . Every iterated monodromy group comes together with a naturally defined action on a rooted tree. We present an effective method to compute this action and show how the dynamics of f is related to the group. In particular, the Julia set of f can be reconstructed from IMG(f) (from its action on the tree), if f is expanding.

C*-algebras and self-similar groups

Abstract We study Cuntz-Pimsner algebras naturally associated with self-similar groups (like iterated monodromy groups of expanding dynamical systems). In particular, we show how to reconstruct the Julia set of an expanding map from the Cuntz-Pimsner algebra of the associated iterated monodromy group and the gauge action on it. We compute K-theory of algebras associated with complex hyperbolic rational functions. It is proved that under some natural conditions the Cuntz-Pimsner algebra of a self-similar group is purely infinite, simple and nuclear. We also show a relation of our algebras with Ruelle algebras of the associated solenoids.

CONJUGATION IN TREE AUTOMORPHISM GROUPS

It is given a full description of conjugacy classes in the automorphism group of the locally finite tree and of a rooted tree. They are characterized by their types (a labeled rooted trees) similar to the cyclical types of permutations. We discuss separately the case of a level homogenous tree, i.e. conjugality in wreath products of infinite sequences of symmetric groups. It is proved those automorphism groups of rooted and homogenous non-rooted trees are ambivalent.

Rigidity of Branch Groups Acting on Rooted Trees

Automorphisms of groups acting faithfully on rooted trees are studied. We find conditions under which every automorphism of such a group is induced by a conjugation from the full automorphism group of the rooted tree. These results are applied to known examples such as Grigorchuk groups, Gupta–Sidki group, etc.

Extensions of amenable groups by recurrent groupoids

We show that the amenability of a group acting by homeomorphisms can be deduced from a certain local property of the action and recurrency of the orbital Schreier graphs. This applies to a wide class of groups, the amenability of which was an open problem, as well as unifies many known examples to one general proof. In particular, this includes Grigorchuk’s group, Basilica group, group associated to Fibonacci tiling, the topological full groups of Cantor minimal systems, groups acting on rooted trees by bounded automorphisms, groups generated by finite automata of linear activity growth, and groups naturally appearing in holomorphic dynamics.

Scale-invariant groups

Motivated by the renormalization method in statistical physics, Itai Benjamini defined a finitely generated infinite group G to be scale-invariant if there is a nested sequence of finite index subgroups G_n that are all isomorphic to G and whose intersection is a finite group. He conjectured that every scale-invariant group has polynomial growth, hence is virtually nilpotent. We disprove his conjecture by showing that the following groups (mostly finite-state self-similar groups) are scale-invariant: the lamplighter groups F\wr\Z, where F is any finite Abelian group; the solvable Baumslag-Solitar groups BS(1,m); the affine groups A\ltimes\Z^d, for any A\leq GL(\Z,d). However, the conjecture remains open with some natural stronger notions of scale-invariance for groups and transitive graphs. We construct scale-invariant tilings of certain Cayley graphs of the discrete Heisenberg group, whose existence is not immediate just from the scale-invariance of the group. We also note that torsion-free non-elementary hyperbolic groups are not scale-invariant.

On a family of Schreier graphs of intermediate growth associated with a self-similar group

For every infinite sequence ω = x 1 x 2 ? , with x i ? { 0 , 1 } , we construct an infinite 4-regular graph X ω . These graphs are precisely the Schreier graphs of the action of a certain self-similar group on the space { 0 , 1 } ∞ . We solve the isomorphism and local isomorphism problems for these graphs, and determine their automorphism groups. Finally, we prove that all graphs X ω have intermediate growth.