Rational subsets of groups

This chapter is devoted to the study of rational subsets of groups, with particular em-phasis on the automata-theoretic approach to finitely generated subgroups of free groups. Indeed,Stallings’ construction, associating a finite inverse automaton with every such subgroup, inaugu-rated a complete rewriting of free group algorithmics, with connections to other fields such astopology or dynamics.Another important vector in the chapter is the fundamental Benois’ Theorem, characterizingrational subsets of free groups. The theorem and its consequences really explain why languagetheory can be successfully applied to the study of free groups. Rational subsets of (free) groups canplay a major role in proving statements (a priori unrelated to the notion of rationality) by induction.The chapter also includes related results for more general classes of groups, such as virtually freegroups or graph groups.