Abstract

. Recently, group theorists have been exploiting finite state automata as a tool in the study of groups. 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 for a pseudovariety of groups H, is it true that a J-extension of a group in H must divide a semidirect product of a J-trivial monoid and a group in H. We show the answer is affirmative if H is closed under extension and may be negative otherwise...