On various profinite completions of groups acting on rooted trees

Abstract: Groups that act faithfully on rooted trees can be studied via their finite quotients. There are several natural collections of finite quotients that can be chosen for this. The mathematical object that encodes all these finite quotients and the maps between them is the profinite completion of the group (with respect to the chosen collection). Taking all possible finite quotients of the group gives *the* profinite completion of the group, annd this maps onto each of the other completions. Determining the kernels of these maps is known as the congruence subgroup problem.  This has been studied by various authors over the last few years, most notably for self-similar groups and (weakly) branch groups. In the case of self-similar regular branch groups, much insight can be gained into this problem using a symbolic-dynamical point of view. After reviewing the problem and previous work on it, I will report on work in progress with Zoran Sunic on determining the dynamical complexity of these completions and calculating some of these kernels with relative ease.

Examples will be given. No previous knowledge of profinite, self-similar or branch groups is required.

group theory

Audience: researchers in the topic

GOThIC - Ischia Online Group Theory Conference

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