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)

Seminar February 23rd

Monday, February 23rd, 13:00, CRM Small Room

Speaker: Borja de Balle Pigem, UPC

Title: Extensions de l'algorisme clàssic de Whitehead.

Seminar February 16th

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

Speaker: Xaro Soler Escrivà, Universitat d'Alacant

Title: Algunes qüestions de permutabilitat en grups finits.

Seminar February 9th

Monday, February 9th, 13:00, CRM Small Room

Speaker: Pedro V. Silva, University of Porto

Title: The mixed orbit problem for the free group of rank 2

Abstract: In 1936, Whitehead solved the orbit problem for words of a free group: given $u,v \in F$, it is decidable whether or not $v =\varphi(u)$ for some $\varphi \in \mbox{Aut} F$. In 1984, Gersten proved a similar theorem for f.g. subgroups of a free group. In both cases, the proofs rely heavily on the use of Whitehead automorphisms to compute shortest words or "smallest" subgroups in the orbit. Such techniques cannot be applied to the mixed orbit problem: given $u \in F$ and $H \leq_{f.g.} F$, is it decidable whether or not $u \in \varphi(H)$ for some $\varphi \in \mbox{Aut} F$? We solve this problem for free groups of rank 2 for the full automorphic orbit and for various subgroups of $\mbox{Aut} F$.

Our techniques involve an appropriate decomposition of $\mbox{Aut} F$ and the study of the dynamical behaviour of a Stallings automaton when we apply automorphisms accordingly. Introducing new concepts such as sources, sinks and singularities for such automata, and proving the existence of certain invariants, we show that the dynamics of such a process is eventually expanding. This fact will lead to the desired decidability results.

These results were obtained in collaboration with Pascal Weil (University of Bordeaux).

Seminar February 2nd

Monday, February 2th, 13:00, CRM Small Room

Speaker: Francesco Matucci, CRM

Title: Structure Theorems for Subgroups of Homeomorphisms Groups.

Abstract: We give a classification of the solvable subgroups G of the group Homeo_+(S^1) of all orientation-preserving homeomorphisms of the unit circle. The key tool is proving that the rotation number map is a group homomorphism and it is done by relating the dynamics of G and its group structure. Applications include new proofs of known results as the Margulis' theorem on the existence of a G-invariant probability measure on S^1 and Burslem-Wilkinson theorem on the classification of solvable groups of analytic diffeomorphisms (this is joint work with C. Bleak and M. Kassabov).

Seminar January 26th

Monday, January 26th, 13:00, CRM Small Room

Speaker: Bob Gilman, Stevens Institute of Technology

Title: Group Theoretical Questions from Cryptography

Abstract: The recent development of cryptosystems based on finitely presented groups has produced a number new questions in combinatorial group theory. We survey these questions and solve one of them.