The main problem considered in this talk is the problem of determining, given a rational subset of a certain group, whether or not this subset is recognizable. In a 1996 paper, Géraud Sénizergues answered this question for free groups in the process of proving a conjecture proposed by Jacques Sakarovitch in 1979:   every rational subset of the free group must be either recognizable or disjunctive (the corresponding syntactical congruence is the identity). We developed two alternative approaches to Sénizergues's that allow us to obtain various algorithmic characterizations of recognizability as well as alternative proofs of Sénizergues's results, including the conjecture of Sakarovitch.

The first approach uses techniques inspired by the study of inverse monoids and inverse automata and provides a wide variety of characterizations, sheding further light on recognizability in this setting.

The second approach is more universal and allows generalizations to other classes of groups, namely to free Abelian groups and also to the general case of finite extensions. As a consequence, we are able to provide a simple and elegant proof for Sakarovitch's conjecture.

Other decidability problems related to rational subsets of groups will be also discussed.