Fribourg Weekend in Group Theory
18 - 20 May 2012
Department of Mathematics,
University of Fribourg,
Switzerland
Organizer: Laura Ciobanu, University of Fribourg (laura.ciobanu@unifr.ch)
Sponsors:
Marie Curie Reintegration Grant 230889; University of Fribourg, Mathematics Department
Speakers
| Gerhard Rosenberger (Hamburg) |
Enric Ventura (Barcelona) |
Armando Martino (Southampton) |
Ian Chiswell (London) |
Markus Lohrey (Leipzig) |
Chloe Perin (Strasbourg) |
| Nic Touikan (Marseille) |
Yago Antolin (Southampton) |
Ben Fine (Fairfield) |
| Alain Valette (Neuchatel) |
Laura Ciobanu (Fribourg) |
Cornelius Reinfeldt (Kiel) |
Schedule of Talks
Titles and Abstracts
Gerhard Rosenberger: Test Elements, Generic Elements and Almost Primitivity in Free Products
B. Fine, G. Rosenberger, D. Spellman and M. Stille examined in
[Pacific J. Math.
190, 1999, 277-297] the relationships between test words, generic elements,
almost primitivity and tame almost primitivity in free groups. In a
common work with A.-K. Engel and B. Fine we extend the concepts and connections to general free products and in particular to free products of cyclic groups.
Yago Antolin: Geodesic growth in right-angled and even Coxeter groups
It has long been known that the spherical or standard growth of a
right-angled Coxeter (or Artin) group depends only on the f-polynomial
of the graph it is based on. Thus there are many non-isomorphic right-
angled Coxeter (or Artin) groups with the same spherical growth.
In this talk we consider geodesic instead of spherical growth, and
discuss which combinatorial properties of a regular graph can
completely determine the geodesic growth of the right-angled Coxeter
group this graph defines. As a consequence, we provide the first known
examples of right-angled and even Coxeter groups with the same
geodesic growth series. This is joint work with Laura Ciobanu.
Markus Lohrey: Rational subsets in groups
Let G be a group with a symmetric generating set A.
A subset of G is rational if it is the homomorphic image of
a rational subset of the free monoid generated by A. Alternatively,
the class of rational subsets can be obtained as the smallest class
of subsets of G that contains all finite subsets and that is closed under
union, product, and Kleene star (= taking the submonoid generated by
a given set). Finitely generated subgroups and finitely generated submonoids
are examples of rational subsets.
A classical result of Benois states that membership in rational subsets
of a free group is decidable. In the talk, I will give a survey on groups,
where membership in rational subsets is (un)decidable.
It turns out that the membership problem for rational subset is closely
related
to the membership problem for finitely generated submonoids. E.g., in
a group with at least two ends, either both problems are decidable or both
problems are undecidable. It is open, whether this holds also for one-ended
groups.
(joint work with Benjamin Steinberg)
Nic Touikan: Hierarchical accessibility of (relatively) hyperbolic groups
In group theory we can think of a graph of groups
decomposition of a group G as "cutting" the group G along the edge
groups. The vertex groups correspond to the resulting pieces. A
hierarchical deomposition of G is a rooted tree first obtained by
"cutting" up G as a graph of groups, then cutting up the resulting
vertex groups, and repeating, repeating...
In 2000 Delzant and Potyagailo gave a proof that every finitely
presented group without 2-torsion admitted a finite hierarchy over the
class of so-called elementary subgroups. Their proof is really nice,
but contains a gap. Lars and myself were able to fix the gap in the
case of torsion-free relatively hyperbolic groups, moreover as a
consequence of JSJ theory (which wasn't as popular back in the late
90's) we are able to obtain a uniform bound on the height of any
elementary hierarchical decomposition of a torsion-free relatively
hyperbolic group. (joint with Lars Louder)
Ben Fine: Something for nothing: Some Consequences of the Solution to the Tarski Problem
Alain Valette: Graphs of groups and the Haagerup property
A group has the Haagerup property if it admits a proper isometric action on a Hilbert space. It is an open question to find a criterion for the Haagerup property for groups acting on trees, assuming that the vertex-stabilizers do have it. We find such a criterion under the assumption that the action on the tree is co-compact, and all vertex- and edge-stabilizers are isomorphic to the same $\Z^n$. In particular, for $n=1$ (generalized Baumslag-Solitar groups), the group has the Haagerup property. This is a joint work with Yves Cornulier.
Enric Ventura: Finite automata for Schreier graphs of virtually free groups
The Stallings construction for finitely generated subgroups of free groups is generalized by introducing the concept of Stallings section, which allows an eficient computation of the core of a Schreier graph based on edge folding. It is proved that those groups admitting Stallings sections are precisely finitely generated virtually free groups, through a constructive approach based on Basse-Serre theory. Complexity issues and applications are also discussed. This is joint work with P. Silva and X. Soler-Escrivà).
Chloe Perin: Forking independence in the free group
Model theorists define, in structures whose first-order theory is "stable" (i.e. suitably nice), a notion of independence between elements. This notion coincides for example with linear independence when the structure considered is a vector space, and with algebraic independence when it is an algebraically closed field. Sela showed that the theory of the free group is stable. In a joint work with Rizos Sklinos, we give an interpretation of this model theoretic notion of independence in the free group using JSJ decompositions.
Ian Chiswell: Ordering graph products of groups
A proof will be outlined that a graph product of (two-sided) orderable
groups is orderable. To do this, a canonical method is needed to order
the free product of a family of groups, given a total order on the index
set of the family. The corresponding result for graph products of
right-orderable groups is much easier to prove.
Cornelius Reinfeldt: Embedding limit groups in NTQ groups - the hyperbolic case
An NTQ-system of equations (non-degenerate triangular quasi-quadratic)
is a system of equations over a free group which is built up of
subsystems of certain very simple types. An NTQ-group is the coordinate
group of an NTQ-system. These groups are a valuable tool for solving
systems of equations over free groups. O. Kharlampovich and A. Miasnikov
have presented an algorithm to embedd a given fully residually free
group (i.e., limit group) into an NTQ group.
In my talk I will describe a generalization of this matter, which is
based on a joint work (in progress) with Olga Kharlampovich. I will give
a brief introduction to limit groups over hyperbolic groups (with
torsion), present a natural generalization of NTQ groups in this setup,
and construct an embedding of a limit group over hyperbolic groups into
an NTQ group.
Armando Martino: Hanna Neumann for Free Products
The Hanna Neumann conjecture has been a long standing conjecture for free groups and was recently solved using ingenious methods combining orderability and analysis by Igor Mineyev. We shall present a version of this proof due to Warren Dicks, but generalised for free products where one uses Kurosh rank instead of the usual rank. However, the original result for free groups remains a special case. This is joint work with Yago Antolin and Inga Schwabrow.
Schedule
The arrival day is Friday, May 18, before 2pm. The departure time is Sunday, May 20, after 1pm. There will be a conference dinner on Saturday evening.
Hotel and conference room
Rooms have been booked for all participants at Hôtel du Faucon, which is a 10 minute walk from the train station and very easy to find (directions).
The talks will take place in the Physics building (see map), room 0.51, which is a 25 minute walk from your hotel, and here are the directions.
Food
Breakfast is not included, but you can purchase it at the hotel, and I highly recommend the confiserie/bakery in front of your hotel.
There are many restaurants close to your hotel. I particularly like the Creperie SucreSale (rue de Lausanne 50). The Vietman House (rue Saint-Michel 3), Restaurant du Midi (rue de Romont 25, Swiss food, service not super friendly), du Gothard (rue du Pont-Mure 16, Swiss food, friendly) Punkt (Place de Notre-Dame 4, Thai and Indian) are listed in increasing price range and are reasonably good. However, the prices will be a shock no matter where you go.
How to get here
If you are coming by train, you can see check out the
Swiss Railroad
pages.
If you are arriving by plane, the Zurich and Geneva airports are equally far from Fribourg, and there are direct trains from the airport to Fribourg every hour.
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