# Seminar: "The rank of subgroups of free groups under generator transformations" (A. Rosenmann, Austrian Institute of Technology)

**Date**: Tuesday November 25th, 13-14.**Place**: CRM, room C1/028**TÃtle**: "The rank of subgroups of free groups under generator transformations".**Speaker**: Amnon Rosenmann (Austrian Institute of Technology).

**Abstract**: We will examine in the talk transformations applied to generators of a finitely-generated subgroup H in a free group F and the effect they have on the rank of H. When the transformations are "dependence transformations" and the transformations graph is without cycles then the rank does not decrease. The "dependence subgroup" of H, Dep H, is the subgroup H < Dep H < F which is generated by all the group elements that depend on H. The rank of Dep H and of the "dependence closure" is at most the rank of H. A special case is the root and the root closure of a subgroup. The results are applied to the inner rank of a system of equations in free groups and in particular to Lyndon Equation. We will examine also the rank of the intersection of subgroups in free groups with respect to transformations on the generators. As a consequence we get that echelon subgroups are inert.