Seminar March 30th

Monday, March 30th, 13:00, CRM Small Room

Speaker: Elena Pribavkina (Ural State University, Ekaterinburg)

Title: A dynamical approach to Thompson's group V

Abstract: A word w over a finite alphabet A is called n-collapsing if for an arbitrary deterministic finite automaton the inequality |\delta(Q,w)|\le |Q| − n holds provided that |\delta(Q,u)|\le |Q| − n for some word u (depending on the automaton). In case n=2 there is a gruop-theoretic characterization of 2-collapsing words. To every word w is associated a finite family of finitely generated subgroups in finitely generated free groups. Then the word is 2-collapsing iff each of these subgroups has index at most 2 in the corresponding free group. There is also a similar characterization for the closely related class of so-called 2-synchronizing words.

Such a characterization allows to establish some new properties of the language of 2-collapsing words, for instance, to show that this language over a binary alphabet is not context free, and to find a new lower bound on the length of 2-collapsing words. In the talk will also be discussed some open problems concerning collapsing words for which such a group-theoretic approach might be useful.

Seminar March 16th

Monday, March 16th, 13:00, CRM Small Room

Speaker: Eugenia Sapir, ENS-Lyon

Title: A dynamical approach to Thompson's group V

Abstract: We describe the structure of centralizers of elements in Thompson's group V. In this group, centralizers are determined by the dynamics of the elements. Each element divides the Cantor set into regions where it acts as the identity, as a permutation of intervals, or as a network between attractors and repellers. We work with two equivalent element representations. The first one is the classical tree pair approach which allows us to use the revealing pair technique introduced by Matt Brin, to easily identify the three types of regions. The second one involves representing elements as diagrams of strands to understand conjugacy classes. (This is joint work with C.Bleak, A.Gordon, G.Graham, J.Hughes, F.Matucci and H.Newfield-Plunkett)