Probabilistic identities, analytic groups and free constructions

Profinite groups can be endowed with a probability measure, which allows one to investigate probabilistic questions in this class of topological groups. A particularly interesting instance is that of a group with a probabilistic identity, that is, when a group satisfies a word with positive probability. In this presentation, I will talk about the structural implications of satisfying a probabilistic identity in two classes of groups: analytic groups and fundamental groups of graphs of pro-p groups. This is joint work with Steffen Kionke, Tommaso Toti, Matteo Vannacci and Thomas Weigel.