Seminar May 11th

Monday, May 11th, 13:00, CRM Small Room

Speaker: Arye Juhasz (Technion, Israel Institute of Technology)

Title: Solution of the Membership Problem for Magnus subsemigroups in certain one-relator groups.

Abstract: Let G be a one-relator group given by a finite presentation . A subgroup of G generated by a proper subset of X is called a Magnus subgroup of G, after W.Magnus, who introduced them to classical Combinatorial Group Theory. They are important building blocks of the theory of one-relator groups. Two of their main properties are that they are free and they have solvable Membership Problem. (The Membership Problem ask for an algorithm to decide whether a given element of the group belongs to a given subgroup).

Quite recently it became clear that the Word Problem for some one-relator algebraic structures which are more complicated than groups can be reduced to the Membership Problem for certain subsemigroups of one-relator groups.

Unfortunately, the existing algebraic and geometrical methods applied to subgroups do not work for subsemigroups. In this talk, using combinatorial methods, we consider the Membership Problem and the Freeness Problem for Magnus subsemigroups of one-relator groups with a small-cancellation condition. (These are subsemigroups of G generated by proper subsets of X union X-1). The main ingredients are van Kampen diagrams and word combinatorics.