Title: Profinite groups of finite probabilistic virtual rank

Abstract: A profinite group G carries naturally the structure of a probability space, namely with respect to its normalised Haar measure.  We say that G has finite probabilistic virtual rank, if there exists a positive integer k such that the closed subgroup generated by k independent random elements of G is almost surely open (and thus of finite index) in G.  In my talk I will discuss some key theorems and open problems about random generation in profinite groups, with a view toward finite direct products of hereditarily just infinite profinite groups.  Classic examples of the latter type of groups are semisimple algebraic groups over non-archimedean local fields.  Then I will discuss selected results from an ongoing joint project with D. Veronelli and M. Vannacci.