Combinatorial group theory, inverse monoids, automata, and global semigroup theory

• Type: Article in International Scientific Journal
• Date: 2002

Authors

delgado, m (Author) FCUP; margolis, s (Author) Other; steinberg, b (Author) Other

Journal

Title: International Journal of Algebra and Computation

Vol. 12

Pages: 179-211

ISSN: 0218-1967

Publisher: World Scientific

Indexing

ISI Web of Knowledge - 10 Citations

Scopus - 9 Citations

Scientific classification

FOS: Natural sciences > Mathematics

Other information

Authenticus ID: P-000-QER

DOI: 10.1142/s0218196702000924

Abstract (EN): This paper explores various connections between combinatorial group theory, semigroup theory, and formal language theory. Let G = [A/R] be a group presentation and B(A,R) its standard 2-complex. Suppose X is a 2-complex with a morphism to B(A,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 AI(A, R) to the presentation (AIR) with the property that pointed subgraphs of covers Of SA,R are classified by closed inverse submoncids of AI(A, R). In particular, we obtain an inverse monoid theoretic condition for a subgroup to he 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.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 33