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 to bi-orderability criterion based on the Alexander polynomial.

Consider a torus bundle over the circle with one boundary. Perron-Rolfsen shows that having an Alexander polynomial with real positive roots is a sufficient condition for a surface bundle with one boundary to have bi-orderable fundamental group. This is done by showing the action induced by the monodromy preserves a “standard” bi-ordering of the fundamental group of the surface. In this talk, we discuss if there are other ways to bi-order the fundamental group of a torus bundle with one boundary component. This work is joint with Henry Segerman. This work is partially funded by NSF grant DMS-2213213.

I will discuss some problems concerning finitely generated (and more generally countable) subgroups of homeomorphism groups with specified properties. In dimension one, I will concentrate on regularity, and discuss some aspects of joint work with S. Kim on critical regularity. I will then discuss the outline of a program to generalize results known in dimension one to higher dimension. Along the way, I will discuss a recent result joint with S. Kim and J. de la Nuez Gonzalez about the model theory of homeomorphism groups of compact manifolds, and indicate how those ideas may be generalized in the future.

Let M be a closed, orientable, and irreducible 3-manifold with Heegaard genus two. We prove that if the fundamental group of M is left-orderable then M admits a co-orientable taut foliation.

A distortion element of a group is an element which is distorted inside a finitely generated subgroup, which means that the word-length of its powers grow sublinearly. Thus talk is inspired in the question whether there is a finitely-generated order able group all of whose elements are distorted. We will provide concrete results in smooth settings.

I will report on the joint works with J. Brum, N. Matte-Bon and M. Triestino about R-focal actions of groups on the real line by homeomorphisms. I will describe and motivate this notion, provide some examples and give some applications to the problem of understanding the space of actions of a given (class of) group(s).

In joint work with Eiko Kin, we consider minimal-volume cusped orientable hyperbolic 3-manifolds. We find that for one cusp, there are two minimal examples: one has bi-orderable fundamental group and the other does not. There is a similar strory for two cusps. To study that case, and others, we consider braids which are “order-preserving,” meaning that the corresponding Artin action on the automorphism group of the free group F_n preserves some bi-ordering of the group.

A context-free ordering on a finitely generated group G is a left ordering on G whose positive cone can be represented by a context-free language (a language recognized by a push-down automaton). We provide constructions of three Cantor sets of orderings on the free group F_k of finite rank k, each of which has a dense subset consisting of context-free orders. The first Cantor set consist of orderings extending the usual lexicographic ordering on the positive submonoid, the second one consists of orderings extending the short-lex ordering, and the third one consists of discrete orderings. The results can be extended to free products in which the free factors admit regular orderings. In particular, we have explicit constructions of orderings on A*B in which A is a convex subgroup.

Given a finitely generated group G, we want to describe all possible actions of G on the line. The space of all such actions, up to pointed semi-conjugacy, form a compact lamination, that can be realized as a subspace of the spaces of actions of G on the line, called the Deroin space of G. I will discuss some of the examples for which we are able to describe such a space. This is part of a series of works in collaboration with Brum, Matte Bon, and Rivas.