COMBINATORIAL GROUP THEORY, INVERSE MONOIDS, AUTOMATA, AND GLOBAL SEMIGROUP THEORY

Abstract

This paper explores various connections between combinatorial group theory, semigroup theory, and formal language theory. Let G = <A|R> be a group presentation and ℬA, R its standard 2-complex. Suppose X is a 2-complex with a morphism to ℬA, R which restricts to an immersion on the 1-skeleton. Then we associate an inverse monoid to X which algebraically encodes topological properties of the morphism. Applications are given to separability properties of groups. We also associate an inverse monoid M(A, R) to the presentation <A|R> with the property that pointed subgraphs of covers of ℬA, R are classified by closed inverse submonoids of M(A, R). In particular, we obtain an inverse monoid theoretic condition for a subgroup to be quasiconvex allowing semigroup theoretic variants on the usual proofs that the intersection of such subgroups is quasiconvex and that such subgroups are finitely generated. Generalizations are given to non-geodesic combings. We also obtain a formal language theoretic equivalence to quasiconvexity which holds even for groups which are not hyperbolic. Finally, we illustrate some applications of separability properties of relatively free groups to finite semigroup theory. In particular, we can deduce the decidability of various semidirect and Mal/cev products of pseudovarieties of monoids with equational pseudovarieties of nilpotent groups and with the pseudovariety of metabelian groups.