Image reducing words and subgroups of free groups

A word w over a *nite alphabet is said to be n-collapsing if for an arbitrary *nite automaton A = 〈Q; −·−〉, the inequality |Q ·w|6 |Q| − n holds provided that |Q · u|6 |Q| − n for some word u (depending on A). We give an algorithm to test whether a word is 2-collapsing. To this aim we associate to every word w a *nite family of *nitely generated subgroups in *nitely generated free groups and prove that the property of being 2-collapsing re0ects in the property that each of these subgroups has index at most 2 in the corresponding free group. We also *nd a similar characterization for the closely related class of so-called 2-synchronizing words. c © 2003 Elsevier B.V. All rights reserved.