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.

Fourth Barcelona Weekend in Group Theory, April 24th and 25th

The program of the Fourth Barcelona Weekend in Group Theory is:

Friday April 24th, Auditori Centre de Recerca Matemàtica

• 15:00 Gilbert Levitt (Université de Caen, France)
  From Baumslag-Solitar groups to companion matrices.


• 16:15 Claas Röver (National University of Ireland, Galway)
  Groups with Context-Free Conjugacy Problem.

• 17:30 Armando Martino (University of Southampton, England)
  Isometries of Outer Space.

Saturday April 25th, FME Aula 102, Edifici U, UPC Campus Sud

• 10:00 Oleg Bogopolski (Universität Düsseldorf, Germany)
  On the fundamental and the first singular homology groups of the Hawaiian Earrings and of Griffiths' space.

• 11:15 Adolfo Ballester Bolinches (Universitat de València, Spain)
  On abnormal maximal subgroups of finite groups.


• 12:30 Goulnara Arzhantseva (Université of Genéve, Switzerland)
  Geometric small cancellation conditions with applications.

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).