Tuesday, October 20th, 15:30, CRM Small Room
Speaker: Olivier Siegenthaler (ETH Zurich)
Title: The Twisted Twin of the Grigorchuk Group
Tuesday
Seminar October 13th
Tuesday, October 13th, 15:30, CRM Small Room
Speaker: Enric Ventura (UPC)
Title: Computing fixed closures in free groups
Abstract: Let F be a finitely generated free group. We present an algorithm such that, given a subgroup H in F, decides whether H is the fixed subgroup of some family of automorphisms, or family of endomorphisms of F and, in the affirmative case, finds such a family. The algorithm combines both combinatorial and geometric methods.
Speaker: Enric Ventura (UPC)
Title: Computing fixed closures in free groups
Abstract: Let F be a finitely generated free group. We present an algorithm such that, given a subgroup H in F, decides whether H is the fixed subgroup of some family of automorphisms, or family of endomorphisms of F and, in the affirmative case, finds such a family. The algorithm combines both combinatorial and geometric methods.
Seminar September 22nd
Welcome to the Barcelona Group Theory Seminar for the year 2009-10. Official meeting time this semester is Tuesdays at 15:30.
Tuesday, September 22nd, 15:30, CRM Small Room
Speaker: Warren Dicks (UAB)
Title: On Leighton's graph covering theorem
Tuesday, September 22nd, 15:30, CRM Small Room
Speaker: Warren Dicks (UAB)
Title: On Leighton's graph covering theorem
Wednesday
Seminar June 29th
Monday, June 29th, 13:00, CRM Small Room
Speaker: Martin Kassabov, Cornell University
Title: Subspace arrangements and property T
Abstract: I will mainly talk about (my viewpoint at) a method for proving property T started by Dymara and Januszkiewicz. Their original motivation came from groups action on dimensional building, but the refined idea does not used anything more angles between subspaces in an finite dimensional Euclidean space. Parts of the talk are based on a work of M. Ershov and A. Jaikin.
Speaker: Martin Kassabov, Cornell University
Title: Subspace arrangements and property T
Abstract: I will mainly talk about (my viewpoint at) a method for proving property T started by Dymara and Januszkiewicz. Their original motivation came from groups action on dimensional building, but the refined idea does not used anything more angles between subspaces in an finite dimensional Euclidean space. Parts of the talk are based on a work of M. Ershov and A. Jaikin.
Friday
Seminar June 3rd
We will have a double session of the seminar at the same time as the Hyperbolic Groups course:
Wednesday June 3rd, Room 005, FME, Campus Sud UPC
First Talk, 11:00
Speaker: José Manuel Higes López, Universidad Complutense de Madrid
Title: The coarse classification of countable abelian groups
Abstract: We will show the classification of countable abelian groups up to coarse equivalence. As an application we will use this result to detect which groups are coarsely equivalent to countable abelian groups. Joint with T. Banakh, I. Zarichnyi.
Second talk, 12:00
Speaker: Lars Louder, University of Michigan
Title: Graphs of subgroups of free groups
Abstract: We will develop, using Stallings' folding, graphs of groups, and a process called "blowing up," a simple topological model for representing graphs of subgroups of free groups. Applications to be discussed include simple proofs of some old results of combinatorial group theory. Generalizations of the technique and of these seemingly simple ideas to limit groups are important in the dimension theory for algebraic varieties defined over the free group.
Wednesday June 3rd, Room 005, FME, Campus Sud UPC
First Talk, 11:00
Speaker: José Manuel Higes López, Universidad Complutense de Madrid
Title: The coarse classification of countable abelian groups
Abstract: We will show the classification of countable abelian groups up to coarse equivalence. As an application we will use this result to detect which groups are coarsely equivalent to countable abelian groups. Joint with T. Banakh, I. Zarichnyi.
Second talk, 12:00
Speaker: Lars Louder, University of Michigan
Title: Graphs of subgroups of free groups
Abstract: We will develop, using Stallings' folding, graphs of groups, and a process called "blowing up," a simple topological model for representing graphs of subgroups of free groups. Applications to be discussed include simple proofs of some old results of combinatorial group theory. Generalizations of the technique and of these seemingly simple ideas to limit groups are important in the dimension theory for algebraic varieties defined over the free group.
Course on Hyperbolic Groups, June 2nd to 4th
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
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)
Thursday
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.
Speaker: Borja de Balle Pigem, UPC
Title: Extensions de l'algorisme clàssic de Whitehead.
Saturday
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.
Speaker: Xaro Soler Escrivà, Universitat d'Alacant
Title: Algunes qüestions de permutabilitat en grups finits.
Monday
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).
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).
Thursday
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).
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.
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.
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