Free profinite semigroups over pseudovarieties of finite semigroups proved to be crucial for the development of Finite Semigroup Theory since the mid 1980's. However, little is known about their structure when the pseudovariety is "large", like when it consists of all finite semigroups. For this case, the first important advances were made in the last decade, many of them based in a beautiful connection with symbolic dynamics introduced by Almeida and in wreath product techniques applied by Rhodes and Steinberg. The most appealing results concern maximal subgroups of free profinite semigroups.

This talk will be focused in two recent works in which the speaker participated. The first one, co-authored with Almeida, concerns the maximal subgroups defined by minimal symbolic dynamical systems, where concrete presentations for some of them are obtained. Results and a general conjecture based on these presentations are made. The second one, co-authored with Steinberg, is about the proof that the maximal subgroup associated to a non-periodic irreducible sofic dynamical system is a free profinite group of countable rank.