Jorge Farina-Asategui will defend his PhD thesis with the title "Groups acting on rooted trees". The opponent is Professor Laurent Bartholdi from the University of Geneva.

Abstract: This thesis is devoted to the study of different aspects of groups acting on rooted trees and their applications. 

First, we focus on the Hausdorff dimension of groups acting on rooted trees. We develop new tools that lead to the solution of several well-known open problems in the subject due to Abért-Virág, Bartholdi, Grigorchuk, Klopsch and Shalev among others. 

Secondly, we study many aspects of the subgroup structure of these groups: first-order theory, rigidity of actions, profinite topology, virtual retracts, etc. The most remarkable application of the obtained results is the disproval of a well-known conjecture of Boston. 

Then, we study several aspects of random elements and random subgroups of branch groups. We generalize several results of Abért and Virág. 

Lastly, we develop a new ergodic theory for self-similar groups. This has found many unexpected applications, the most remarkable ones being to several problems in arithmetic dynamics. In fact, we use this new theory to obtain a long-awaited classification of the fixed-point proportion of geometric iterated Galois groups of polynomials.