3.30-4.30 pm, Seminar Hall
K. Lakshmanan Memorial Distinguished Lecture
About the lattice of finitely generated subgroups of a free group
LaBRI, CNRS and University of Bordeaux, France.
It is well-known that the subgroups of a free group F are free, but a free group may sit in another in many different ways: e.g. as a finite-index subgroup, as a free factor. We will discuss the lattice of extensions of a finitely generated subgroup H, that is the set of subgroups of F which contain H. Our first consideration will be Takahasi's theorem: a finitely generated subgroup H of F has a finite lattice of extensions, the so-called algebraic extensions, such that every every subgroup containing H is a free multiple of an algebraic extension (a 1951 theorem, with a surprisingly simple automata-theoretic proof). This is to say that a lot of the interesting extensions of H are to be found among its algebraic extensions (here, we consider that being a free multiple is "un-interesting"). For instance, the malnormal or the pure closure of H are algebraic extensions. We will also explore in detail the lattice of finite-index extensions of H, which is a sublattice of the lattice of algebraic extensions.
Along the way, we will pay particular interest to algorithmic questions, concerning the computation of the lattice of algebraic extensions and of the various closures of a given subgroup H, with a particular taste for the situations where classical results and constructions from automata theory can be used.
Many of the results we will present were obtained in joint work with A. Miasnikov, P. Silva and E. Ventura.