Probabilistic identities in profinite groups

• Autores: Tommaso Toti
• Directores de la Tesis: Gustavo Adolfo Fernández Alcober (dir. tes.) , Thomas Weigel (codir. tes.)
• Lectura: En la Universidad del País Vasco - Euskal Herriko Unibertsitatea ( España ) en 2025
• Idioma: inglés
• Programa de doctorado: Programa de Doctorado en Matemáticas y Estadística por la Universidad de La Laguna; la Universidad de Oviedo; la Universidad de Zaragoza; la Universidad del País Vasco/Euskal Herriko Unibertsitatea y la Universidad Pública de Navarra
• Menciones:
  • Internacional
• Calificación: Sobresaliente Cum Laude
• Enlaces
  • Tesis en acceso abierto en: ADDI
• Resumen

  • The thesis deals with the theory of identities on groups from a probabilistic point of view. Given a profinite group 𝐺 and a word 𝑤 , we say that 𝑤 is a probabilistic identity on 𝐺 if it is satisfied with positive Haar measure. Kinds of probabilisitc identities are identities and coset identities, that is, words that are satisfied on a coset of a finite index subgroup. The presence of a probabilistic identity may influence the algebraic structure of the group, for instance it may induce an identity or a coset identity.

    The first two chapters of the thesis focus on introducing to the reader the essential tools used in the research part of the thesis, such as profinite groups, analytic groups, Haar measure, probabilistic identities and randomly free grups. The third chapter is devoted to the study of torsion probabilistic identities. Here, given a prime number 𝑝 , the study of conjugacy classes of torsion elements and Lie methods are used to solve open problems regarding the measure of the torsion elements in various pro-𝑝 groups. An important class covered is that of 𝑝 -adic analytic pro-𝑝 groups, in which also new examples of torsion probabilistic identities are constructed. Chapter 4 focuses on probabilistic Tits alternatives. Given a non-archimedean local field 𝐾 , we deduce a probabilistic Tits alternative for compact groups that are linear over 𝐾 . Then we give a criterion to establish whether a pro-𝑝 group is randomly free in terms of its ℤ𝑝 -representations and we apply it to establish that various classes of pro- 𝑝 groups satisfy probabilistic Tits alternatives. Finally, in Chapter 5 we study a family of finite 𝑝 -groups that arise from pro-𝑝 groups that are not 𝑝 -adic analytic, that is the standard wreath products 𝐶𝑝 ≀ 𝐶𝑝𝑘with 𝑘 ∈ ℕ . For every natural number 𝑛 , we find an upper bound for the probability that a group in this family satisfies the Engel word 𝑒𝑛 and, as an application, we show that 𝑒𝑛 is not a probabilistic identity on the pro-𝑝 wreath product 𝐶𝑝 ≀ ℤ𝑝 .