*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*

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We are looking forward to see you in September.

# Participants

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.

### 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 |
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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 |
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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 |
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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. |
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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 |
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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 |
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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. Lectures 2 and 3: The case of finite first homology. 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. |
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Name |
Bertrand Deroin | Institution | Université Paris-Sud Faculté des Sciences d´Orsay |

Title |
Orderable groups and dynamics |
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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 |
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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. |
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Name |
Andrés Navas | Institution |
Universidad de Santiagode Chile |

Title |
Spaces of left-orderings. |
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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. |
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Name |
Cristóbal Rivas | Institution |
Universidad de Santiagode Chile |

Title |
On the space of left-orderings of virtually solvable groups |
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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. |