Self-similarity (when part of an object is a scaled version of the whole) is one of the most basic forms of symmetry. While known and used since ancient times, its use and investigation took off in the 1980s thanks to the advent of fractals, whose infinite self-similar structure has captured the imagination of mathematicians and lay people alike.

Self-similar fractals are highly symmetrical, so much so that even their symmetry groups exhibit self-similarity. In this talk, I will introduce and discuss groups which are self-similar, or fractal, in an algebraic sense; their connections to fractals, symbolic dynamics and automata theory; how they produce fascinating new examples in group theory, and some research questions in this lively new area.

Hausdorff dimension has become a standard tool to measure the "size" of fractals in real space. However, it can be defined on any metric space and therefore can be used to measure the "size" of subgroups of, say, pro-p groups (with respect to a chosen metric). This line of investigation was started 20 years ago by Barnea and Shalev, who showed that p-adic analytic groups do not have any "fractal" subgroups, and asked whether this characterises them among finitely generated pro-p groups. I will explain what all of this means and report on joint work with Oihana Garaialde and Benjamin Klopsch in which, while trying to solve this problem, we ended up showing an analogue of a theorem of Schreier in the context of pro-p groups of positive rank gradient: any finitely generated infinite normal subgroup of a pro-p group of positive rank gradient is of finite index. I will also explain what "positive rank gradient" means, and why pro-p groups with such a property are "free-like".

Given a group one of the most natural things one can study about it is its subgroup lattice, and the maximal subgroups take a prominent role. If the group is infinite, one can ask whether all maximal subgroups have finite index or whether there are some (and how many) of infinite index. After telling some historical developments on this question, I will motivate the study of maximal subgroups of groups of intermediate growth and report on joint work with Dominik Francoeur where we give a complete description of all maximal subgroups of some "siblings" of Grigorchuk's group.

Groups of rooted tree automorphisms, and (weakly) branch groups in particular, have received considerable attention in the last few decades, due to the examples with unexpected properties that they provide, and their connections to dynamics and automata theory. These groups also showcase interesting phenomena in profinite group theory. I will discuss some of these and other profinite completions that one can use to study these groups, and how to find them. All these concepts will be defined in the talk.