Solved and unsolved problems around one group

Abstract

This is a survey paper on various topics concerning self-similar groups and branch groups with a focus on those notions and problems that are related to a 3-generated torsion 2 group of intermediate growth G, constructed by the author in 1980, and its generalizations Gω, ω ∈ {0, 1, 2}ℕ. Among the topics touched in the paper are: actions on rooted trees and groups generated by finite automata, self-similarity, contracting properties, various notions of length, topology in the space of finitely generated groups, Kolmogorov complexity in relation to algorithmic problems, algorithmic problems such as word, conjugacy, and isomorphism problem, L-presentations of self-similar groups by generators and relators, branch groups and their subgroup structure, maximal and weakly maximal subgroups and the congruence subgroup property for groups acting on rooted trees, groups of finite type, intermediate growth and growth functions, amenability, Schreier graphs associated to groups acting on rooted trees and their asymptotic characteristics, C*-algebras associated to self-similar groups and spectral properties of Markov and Hecke type operators, multidimensional rational maps which arise in the study of the spectral problem, etc. The paper contains many open problems, some of which have long history but some are completely new.