Finite state automata: A geometric approach

Recently, finite state automata, via the advent of hyperbolic and automatic groups, have become a powerful tool in geometric group theory. This paper develops a geometric approach to automata theory, analogous to various techniques used in combinatorial group theory, to solve various problems on the overlap between group theory and monoid theory. For instance, we give a geometric algorithm for computing the closure of a rational language in the profinite topology of a free group. We introduce some geometric notions for automata and show that certain important classes of monoids can be described in terms of the geometry of their Cayley graphs. A long standing open question, to which the answer was only known in the simplest of cases (and even then was non-trivial), is whether it is true, for a pseudovariety of groups $\mathbf{H}$, that a ${\mathcal J}$-trivial co-extension of a group in $\mathbf{H}$ must divide a semidirect product of a ${\mathcal J}$-trivial monoid and a group in $\mathbf{H}$. We show the answer is affirmative if $\mathbf{H}$ is closed under extension, and may be negative otherwise.