The proposer suggests a research plan involving properties of self-similar groups acting on rooted trees, with particular attention paid to classes of self-similar groups enjoying additional finiteness properties such as being generated by a finite self-similar set, being branch groups (having rigid stabilizers of finite index), being finitely constrained (being defined by finitely many forbidden patterns), being contracting (having finite nucleus), being torsion, etc. Among the proposed directions of study are problems involving growth questions, correspondence between notions and results in symbolic dynamics and the theory of self-similar groups, Hausdorff dimension of closures of self-similar groups, questions relating torsion, growth, Hausdorff dimension and the branching property in the setting of self-similar groups, and properties of some particularly interesting groups such as Hanoi Towers groups and the tent map groups.
Broadly speaking, the notion of self-similarity concerns an entity in which many copies of the original can be found at various scales within the entity itself. This is a fundamental notion reflected in many ways both in nature (well known examples include structure of clouds, coastal shapes, plant branching patterns, etc.) and in mathematics (as a recursion, iteration, self-reference, renormalization, etc.). The notion of self-similarity was recently introduced into group theory through actions on rooted trees. The point of view of studying some aspects and some classes of groups through actions on rooted trees has proved to be rather fruitful, since it allows the introduction of many natural(visual) concepts and ideas, while simplifying the notation and presentation. In fact, it is precisely this shift in language that enabled better intuition and resulted in the ongoing explosion of insights, breakthroughs, and links to other areas of mathematics. The current research moves in many directions, reflecting the richness of the subject and its wide appeal and applicability (we can explain and relate in the language of self-similar groups some 2000 years old problems such as Chinese Rings, modern mathematical constructions such as groups of intermediate growth, and entirely new concepts such as iterated monodromy groups).
