Abstracts Orderability

A left-order on a finitely genarated group is regular if its positive cone is the evaluation of a regular language over a finite generating set. In this talk, I will present some examples of left-orders that are regular and some constructions that preserve regularity. This is a joint work with C. Rivas and H.L. Su.

In this talk, I will introduce the notion of orderability of quandles (right and left orderability, right and left circular orderability, and bi-orderability of quandles), and its relation with the notion of orderability of 3-manifold groups. I will also talk about how orderability of quandles can be used to answer questions about orderability of 3-manifold groups.

Let G be a left-orderable group. A research trend in the modern study of orderable groups consists of giving the set of left-orderings of G, denoted by LO(G), the so called Chabauty topology. Whenever G is countable, LO(G) is a compact and completely metrizable topological space. Further, since G acts continuously on LO(G) by conjugacy, we can regard LO(G) as a G-space. This additional feature is fundamental to recent results in the field.

In this talk, I will discuss the ongoing program of investigating left-orderable groups and the corresponding conjugacy action under the viewpoint of Borel complexity theory. Our approach imports techniques from descriptive set theory and shows that the Borel cardinality of the orbit space LO(G)/G can be used to organize left-orderable groups into complexity classes. Time permitting, I will discuss the problem of determining the complexity of left-orderable fundamental groups of 3-manifolds and how it relates to the L-space conjecture.
This is joint work with Adam Clay.

I will survey some of what is known about orderability of 3-manifold groups, especially as it relates to the L-space conjecture which postulates connections between group orderability, taut foliations, and Heegaard Floer homology. A particular focus will be on techniques for showing that particular hyperbolic 3-manifold groups are orderable.

There is a beautiful rigidity result due to Kathryn Mann and Maxime Wolff about circular orderings of the mapping class group of a once punctured finite-type surface. I will talk about two exploitations of their result. The first is a computation of how much the once-punctured mapping class group fails to be left-orderable, that is, a computation of its obstruction spectrum. The second is a result about translation numbers of elements in the mapping class group of a compact surface with one boundary component, and their relation to fractional Dehn twist coefficients. Everything is joint work with Adam Clay.

Using the notion of order-detection of slopes for 3-manifolds with torus boundary due to Boyer and Clay, we describe some results on the left-orderability of the fundamental groups of 3-manifolds that contain incompressible tori. In particular, this gives new evidence in support of the L-space conjecture. This is joint work with Steve Boyer and Ying Hu.

A generalized torsion element is an element such that some product of its conjugates is the identity. A generalized torsion element serves as an (obvious) obstruction for the bi-orderbility of groups. In this talk, we discuss several notions generalized torsion elements, such as, a group whose generalized torsions are torsion, or, a group consisting of generalized torsion elements. We also give an lower bound of the order of generalized torsion elements by using the Alexander polynomial. This is closely related