Abstracts


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.