As an event for the graduate class "Seminar on Group Theory", there will be a three-day minicourse.
Tuesday June 2nd to Thursday June 4th, 15.00 to 18:00, Room 005, FME, UPC Campus Sud
Speaker: Oleg Bogopolski, Universität Düsseldorf
Title: Hyperbolic groups
Thursday
Seminar May 25th
Monday, May 25th, 13:00, CRM Small Room
Speaker: Lluís Bacardit (UAB)
Title: A generalization of theorem of Birman and Hilden
Abstract: A Theorem of Birman and Hilden says that the standard representation of the braid group by automorphisms of a free group $F_n$ remains faithful after passing to a free product of cyclic groups $C_k^{*n}$. If we take a finite index subgroup of $C_k^{*n}$ invariant by the action of the braid group, the action also is faithful. These two facts allow us to embed braid group in mapping class group of genus greater than 0. We will give a generalization of the Birman Hilden theorem to orientable surface with one boundary component and show some inclusion of mapping class groups in mapping class groups of greater genus.
Speaker: Lluís Bacardit (UAB)
Title: A generalization of theorem of Birman and Hilden
Abstract: A Theorem of Birman and Hilden says that the standard representation of the braid group by automorphisms of a free group $F_n$ remains faithful after passing to a free product of cyclic groups $C_k^{*n}$. If we take a finite index subgroup of $C_k^{*n}$ invariant by the action of the braid group, the action also is faithful. These two facts allow us to embed braid group in mapping class group of genus greater than 0. We will give a generalization of the Birman Hilden theorem to orientable surface with one boundary component and show some inclusion of mapping class groups in mapping class groups of greater genus.
Tuesday
Seminar May 18th
Monday, May 18th, 13:00, CRM Small Room
Speaker: Yago Antolín (UAB)
Title: Grupos de dimensión asintótica 1.
Abstract: Veremos ejemplos de grupos de dimensión asintótica 0,1 y 2. Caracterizaremos algebraicamente los grupos finitamente presentados de dimensión asintótica menor o igual que 1. El juego del Hex.
Speaker: Yago Antolín (UAB)
Title: Grupos de dimensión asintótica 1.
Abstract: Veremos ejemplos de grupos de dimensión asintótica 0,1 y 2. Caracterizaremos algebraicamente los grupos finitamente presentados de dimensión asintótica menor o igual que 1. El juego del Hex.
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.
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
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.
Friday
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.
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
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)
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)
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