Orderable groups, 1-5 September 2014, Cajón del Maipo, Chile

Minicourses:

1) Ordered groups and topology.
Adam Clay (U. Québec / Montréal).

2) Orderable group actions and the deep-fall property.
Igor Mineyev (Illinois Univ. at Urbana Champaign).

3) Spaces of orderings,
Andrés Navas (USACH), Cristóbal Rivas (USACH), and Tetsuya Ito (U. Kyoto).

4) Orderable groups and dynamics,
Bertrand Deroin (ENS Paris, CNRS).

Practical information

The conference will be held at the lodge “Cascada de las Ánimas” www.cascadadelasanimas.cl at Cajón del Maipo, a beautiful landscape on the hillside of the Andes Mountains, 1 hour away from Santiago de Chile.

The organizers will take care of all participants transportation from Santiago´s airport (on Sunday 31 August) to the place of the conference and back (on Saturday 06 by noon). The lodging of all participants is also covered by the organization, including meals and coffee breaks from Monday to Saturday morning.

Besides this, the lodge counts with activities like rafting, canopy, horse back riding, massages. Cascada accepts credit cards, but does not have an ATM. Therefore, if you would like to go around and get to know more of Cajón del Maipo, you will need to get some local currency in the airport. Cascada also has a SUV for rent. There is an ATM  in the village of San José de Maipo, which is 10 minutes away by car or public transportation.

Weather in September (the beginning of the spring) in this part of Chile is cold in the mornings (5-10°C) and then the temperature goes up 15-20°C in the afternoon, so bring your jacket and boots, and maybe an umbrella, because there is always the possibility of rain.

In case of problems upon your arrival, you can contact us to the phone number 56 9 74845109.

Last but not least, if you are entering to Chile with a passport issued by Australia, Canada, México, or USA, you have to pay a Reciprocity Tax at the airport; please check: www.aeropuertosantiago.cl/english/index.php?option=com_content&id=35&task=view&Itemid=51

 

We are looking forward to see you in September.

 

Participants

Juan Alonso, Univ. de la República, Uruguay.
Diego Arcis, Univ. de Bourgogne, Dijon, France.
Marcos Barrios, Univ. de la República, Uruguay.
Collin Bleak*, Univ. St Andrews, Scotland.
Michel Boileau*,  Univ. Paul Sabatier, Toulouse, France.
Steven Boyer, Univ. du Québec à Montréal, Canada.
Joaquín Brun, Univ. de la República, Uruguay.
Danny Calegari*, Univ. of Chicago, USA.
Matthieu Calvez, Univ. de Santiago, Chile.
Carolina Canales, Univ. d´Orsay, France.
Gonzalo Castro, Univ. de Santiago, Chile.
Paulina Cecchi, Univ. de Chile.
Adam Clay, Univ. of Manitoba, Canada.
Patrick Dehornoy*, Univ. de Caen, France.
Bertrand Deroin, ENS Paris, France.
Mahdi Teymuri Garakani, Univ. de Santiago, Chile.
Natalia González, Univ. de Chile.
Juan González Meneses, Universidad de Sevilla, Spain.
Cameron Gordon, Univ. of Texas at Austin, USA.
Nancy Guelman, Univ. de la República, Uruguay.
Tetsuya Ito, Univ. of Kyoto, Japan.
Eduardo Jorquera, Pont. Univ. Católica de Valparaíso, Chile.
Dawid Kielak, Univ. Bonn, Germany.
Nicolás Libedinsky, Univ. de Chile.
Lucy Lifschitz*, Univ. of Oklahoma, USA.
Peter Linnell*, Virginia Tech. Blacksburg, USA.
Yash Lodha, Cornell Univ., USA.
Patrizia Longobardi*, Univ. degli studi di Salermo, Italy.
Jérôme Los, Univ. de Provence, France.
Mercede Maj*, Univ. degli studi di Salermo, Italy.
Yoshifumi Matsuda, Aoyama Gakuin University, Japan.
Igor Mineyev, Univ. Illinois at Urbana-Champaigne, USA.
Nicolas Monod*, É. Polytechnique Féd. de Lausanne, Switzerland.
Ignacio Monteverde, Univ. de la República, Uruguay.
Dave Morris, Univ. of Lethbridge, Canada.
Andrés Navas, Univ. de Santiago, Chile.
Pamela Pastén, Univ. de Chile.
Luis Paris, Univ. de Bourgogne, Dijon, France.
José Luis Pérez, Univ. de Santiago, Chile.
Yago Antolín Pichel, Vanderbilt Univ., Nashville, USA.
Fermín Reveles, CIMAT, México.
Cristóbal Rivas, Univ. de Santiago, Chile.
Rachel Roberts, Washington Univ., USA.
Dale Rolfsen*, Univ. British Columbia, Canada.
Zoran Sunic, Texas A&M Univ., College Station, USA.
Alexey Talambutsa, Univ. of Geneva, Switzerland.
Romain Tessera*, Univ. d´Orsay, France.
Aníbal Velozo, Princeton Univ., USA.
Andrea Vera, Univ. de Santiago, Chile.
Alden Walker, Univ. of Chicago, USA.

* = to be confirmed.

For any information about the conference, please send us an email to  ordering.groups@usach.cl

 

 

 

TALKS
Name Yago Antolin Pichel Institution Vanderbilt University
Title  Local indicability and one-relator structures.
Abstract  In this talk I will review some classical theorems about local indicability for one-relator quotients and an approach to them via Bass-Serre Theory.
Name Steven Boyer Institution Université du Québec `a Montréal
Title Foliations, orders, representations, L-spaces and  graph manifolds
Abstract Much work has been devoted in recent years to examining relationships between the existence of a co-oriented taut foliation in a closed, connected, prime 3-manifold W, the left-orderability of the fundamental group of W, and the property that W not be a Heegaard-Floer L-space. When W has a positive first Betti number, each of these conditions holds. If W is a non-hyperbolic geometric manifold the conditions are known to be equivalent. In this talk I will discuss joint work with Adam Clay concerning the case that W is a graph manifold rational homology 3-sphere. We show that W has a left-orderable fundamental group if and ony if it admits a co-oriented taut foliation and show that these conditions imply that W is not an L-space.
Name Cameron Gordon Institution The University of Texas at Austin
Title Left-orderability and cyclic branched covers
Abstract   It is conceivable that for a prime rational homology 3-sphere M, the following conditions are equivalent: (1) pi_1(M) is left-orderable, (2) M admits a co-orientable taut foliation, and (3) M is not a Heegaard Floer homology L-space. We will discuss these properties in the case where M is the cyclic branched cover of a knot. This is joint work with Tye Lidman.
Name Dawid Kielak Institution Universität Bonn
Title Groups with infinitely many ends and fractions
Abstract We will investigate some obstructions of a topological nature which prohibit a group from being a fraction group of a finitely generated subsemigroup. We will then apply our investigation to free groups and obtain two applications: we will see that free groups do not admit isolated orderings nor finite Garside structures.
Name Yash Lodha Institution Cornell University, USA
Title A geometric solution to the von Neumann-Day problem for finitely presented groups.
Abstract We will describe a finitely presented group of homeomorphisms of the circle that is non-amenable and does not contain non-abelian free subgroups.
Name Patrizia Longobardi Institution Università degli studi di Salerno
Title Some results on small doubling in ordered groups
Abstract A finite subset S of a group G is said to satisfy the small doubling
property if |S^2| ≤ α|S| + β, where α and β denote real numbers, α > 1 and
S^2 = {s1s2 | s1, s2 ∈ S}.
Our aim in this talk is to investigate the structure of finite subsets S
of orderable groups satisfying the small doubling property with α = 3 and
small β’s, and also the structure of the subgroup generated by S. This is
a step in a program to extend the classical Freiman’s inverse theorems (see
[?]) to nonabelian groups.
Let G be an orderable group and let S be a finite subset of G of size
|S| = k ≥ 2. We proved in [?] that if |S| > 2 and |S^2| ≤ 3|S| − 4, then S is
a subset of an abelian geometric progression. Moreover, if |S^2| ≤ 3|S| − 3,
then {S} is abelian; the result is the best possible, in fact for any k ≥ 2 we
construct an orderable group with a subset S of order k such that
|S^2| =3k − 2 and {S} is not abelian.
In this talk we present some recent results concerning the structure of the
subset S of an ordered group and the structure of {S}, if
|S^2| ≤ 3|S| − 3 + b,      for some integer b ≥ 1.
We prove that if |S| > 3 and |S^2| ≤ 3|S| − 2, then either {S} is abelian
and at most 3-generated, or {S} is 2-generated and one of the following holds:
(i) {S} = {a, b | [a, b] = c, [c, a] = [c, b] = 1},
(ii) {S} is the Baumslag-Solitar group B(1, 2), i.e. {S} = {a, b | a^b = a^2};
(iii) {S} = {a, b | a^b^2= aab = aba},
(iv) {S} = {c} × {a, b | ab = a^2}.
In particular,
{S} is metabelian, and if it is nilpotent, then its nilpotence class is at most 2.
If {S} is abelian and |S^2| ≤ 3k−2, then the set S has Freiman dimension
at most 3, and the precise structure of S follows from some previous results
of G. A. Freiman. We also describe the exact structure of S if |S^2| ≤ 3k − 2
an (ii) or (iii) or (iv) holds.
Name Jérôme Los Institution Université de Provence
Title A formula for volume entropy of classical presentations for all surface groups
Abstract Using dynamical system arguments we prove an explicit formula to compute the volume entropy of all surface groups for the classical presentations.
Name Dave Morris Institution University of Lathbridge
Title Survey of invariant orders on arithmetic groups
Abstract At present, there are more questions than answers about the existence of an invariant order on an arithmetic group.  We will discuss four different versions of the problem: the order may be required to be total, or allowed to be only partial, and the order may be required to be invariant under multiplication on both sides, or only on one side.  One version is trivial, but the other three are related to interesting conjectures in the theory of arithmetic groups.
Name Rachel Roberts Institution Washington University
Title The Li-Roberts Conjecture
Abstract Suppose M is an irreducible, rational homology sphere.
Boyer, Gordon and Watson have made the following conjecture:
$pi_1(M)$ is left orderable if and only if M is not an L-space.
Ozsv´ath and Szab´o have asked whether it is true that
M is not an L-space if and only if M contains a taut oriented foliation.I will describe work, joint with Tao Li, in which we establish the
existence of
taut oriented foliations in manifolds $M_k(s)$ obtained by s Dehn filling
a knot $k$ in $S^3$, for s sufficiently small. It follows that
$pi_1(M_k(1/n))$ is left orderable whenever n is sufficiently large.
Name Zoran Sunic Institution Texas A&M University
Title Ordering free groups and free products.
Abstract We utilize a criterion for the existence of a free subgroup acting freely on at least one of its orbits to construct such actions of the free group on the circle and on the line, leading to orders on free groups that are particularly easy to state and work with.

We then switch to a restatement of the orders in terms of certain quasi-characters of free groups, from which properties of the defined orders may be deduced (some have positive cones that are context-free, some have word reversible cones, some of the orders extend the usual lexicographic order, and so on).

Finally, we construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another. As an application, we provide a short proof of Vinogradov´s result that the free product of left-orderable groups is left-orderable.

Name Alden Walker Institution University of Chicago
Title Transfers of quasimorphisms
Abstract Let F be a free group.  I´ll describe a transfer construction which lifts the rotation number quasimorphism from a finite index subgroup of F to F, and I´ll give a combinatorial explanation of when such a construction can be extremal for a given word in the free group.  This is joint work with Danny Calegari.
MINICOURSES
Name Adam Clay Institution University of Manitoba, Canada
Minicourse Orderable groups and topology
Abstract The goal of this minicourse is to study the orderability properties of fundamental groups of 3-manifolds, and when possible, explain orderability or non-orderability of the fundamental group via topological properties of the manifold.  In particular I will cover bi-orderability of knot groups, connections with foliations, group actions and the L-space conjecture; the lectures will include plenty of open problems and conjectures that are active areas of research. Owing to a theorem of Boyer, Rolfsen and Wiest (to be covered in the first lecture), this material is naturally best organized into two cases:  The case of infinite first homology, and the case when the first homology is finite.  The lectures will therefore cover material as follows:

Lecture 1: The case of infinite first homology.
-The theorem of Boyer, Rolfsen and Wiest, bi-orderability of manifolds that fiber over the circle and knot manifolds.

Lectures 2 and 3: The case of finite first homology.
-A review of Seifert fibered manifolds, and the connection between foliations and left-orderings in the Seifert fibered case.
-The L-space conjecture, its relationships with the operation of Dehn surgery, and the expected behaviour of left-orderability with respect to Dehn surgery.

References: Notes and a list of references will be available for each talk.

Name Igor Mineyev Institution Univ. Illinois at Urbana-Champaigne, USA
Minicourse Orderable group actions and the deep-fall property.
Abstract The above title is intensionally misleading: “orderable” can be either a
group or an action. Orderable actions (on graphs) naturally occurred in
the systems of graphs that were used to prove the Strengthened Hanna
Neumann Conjecture (SHNC). We will discuss generalizations from systems of
graphs to systems of complexes, and from SHNC to submultiplicativity. The
deep-fall property can be defined for orderable actions on graphs, and
also in the general setting. It implies both SHNC and submultiplicativity.
It is therefore an interesting question, which orderable actions have this
property. This is also related to some long-standing questions in operator
algebras and ring theory.
Name Bertrand Deroin Institution Université Paris-Sud Faculté des Sciences d´Orsay
Title Orderable groups and dynamics
Abstract The lectures will focus on the dynamics of countable groups acting faithfully on the real line by preserving orientation homeomorphisms. As is well-known, those groups are precisely the countable groups that admit a left-order. The first part will be dedicated to the study of contraction properties of such actions, with applications to the problem of existence of a free subgroup, and the second will discuss the notion of almost-periodic actions, among them being the interesting harmonic ones. A nice object coming out here is a compact one dimensional foliated space, namely the space of almost-periodic actions (resp. the space of normalized harmonic ones), which can serve as a substitute to the space of left-orders (this latter will be discussed in the mini-course by Navas/Rivas/Ito/Paris). Dynamical properties of this foliation, as for instance the existence of periodic orbits, fixed points, invariant measures etc.. reveal some interesting properties of the algebraic structure of the group, as we will try to explain.
Name Tetsatoya Ito Institution Research Institute for Mathematical Sciences, Kyoto University
Title Constructing isolated orderings
Abstract An isolated ordering, though its definition is easy, is not easy to find
and our catalog of isolated orderings are still unsataisfactory.
In this talk I will review current method about how to get an isolated
orderings:
* Dehornoy-like ordering
* Triangular presentation and word reversing
* Amalgamated products
Instead of giving detailed arguments which are often technical, we
emphasize their background idea (in somewhat informal style). This will
illustrate why isolated orderings are interesting.
Name Andrés Navas Institution Universidad de Santiagode Chile
Title Spaces of left-orderings.
Abstract The space of orders of a group was introduced by Ghys and independently by Sikora. In general, this is a totally disconnected compact space upon which the group acts by conjugacy; moreover, for countable groups, it is metrizable.
In this talk I will speak about some general properties of this space as well several open questions on its structure. In particular, I will describe many of the available proofs of that the space of left-orders of the free group is a Cantor set.
Name Cristóbal Rivas Institution Universidad de Santiagode Chile
Title On the space of left-orderings of virtually solvable groups
Abstract A general strategy for trying to approximate a left-ordering on a group, is to approximate the given ordering by its conjugates. For instance, Navas has shown that this strategy always works unless the Conradian Soul of the initial ordering is a group admitting only finitely many left-orderings.
In this talk, we will review this method and show why in the case of solvable groups this leads to the following dichotomy: the space of left-orderings of a countable solvable group is either finite or a Cantor set.