@phdthesis{718511b0-6347-4ee9-9ed3-2b99bf54c08a,
  abstract     = {{This thesis is devoted to the study of different aspects of groups acting on rooted trees and their applications.<br/><br/>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.<br/><br/>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.<br/><br/>Then, we study several aspects of random elements and random subgroups of branch groups. We generalize several results of Abért and Virág.<br/><br/>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.}},
  author       = {{Fariña Asategui, Jorge}},
  isbn         = {{978-91-8104-933-6}},
  issn         = {{1404-0034}},
  keywords     = {{Hausdorff dimension; Hausdorff spectra; self-similar groups; branch groups; random subgroups; dynamical systems; ergodicity; fixed-point proportion; iterated monodromy groups; iterated Galois groups; polynomials; Hausdorff dimension; Hausdorff spectra; self-similar groups; branch groups; random subgroups; dynamical systems; ergodicity; fixed-point proportion; iterated monodromy groups; iterated Galois groups; polynomials}},
  language     = {{eng}},
  publisher    = {{Lund University}},
  school       = {{Lund University}},
  title        = {{Groups acting on rooted trees : Dimension, subgroups, randomness and dynamics}},
  year         = {{2026}},
}