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# Investigating the L-space conjecture: Via orderability

PSL(2,R) representations and left-orderablility of Dehn filling- Xinghua Gao (KIAS)

Constructing non-trival $\widetilde{PSL}(2, R)$ representations has been proven to be a useful tool for showing the left-orderablity of a 3-manifold group. In this talk, I will show how to use $\widetilde{PSL}(2, R)$ representations of the fundamental group of a knot complement to determine which Dehn filling of it has left-orderable fundamental group. In particular, I will compute slopes of left-orderable Dehn filling of a class of two-bridge knots as an example.

Promoting circular-orderability to left-orderability- Ty Ghaswala (UQAM)

I will present new necessary and sufficient conditions for a circularly-orderable group to be left-orderable, and introduce the obstruction spectrum of a circularly-orderable group. I will then show how newly developed machinery can help us compute the obstruction spectrum in a variety of examples, including 3-manifold groups relevant to the L-space conjecture, and mapping class groups. I will finish the talk with new progress in answering the question of when the direct product of two circularly-orderable groups is circularly-orderable, a fundamental question which is frustratingly difficult to say anything about.

Left-orderability of 3-manifold groups and foliations of $3$-manifolds- Ying Hu (Nebraska)

In this talk, we will discuss how the existence of certain nice dimension 1 and dimension 2 foliations of $3$-manifolds can lead to the left-orderability of their fundamental groups. We will give some applications of these observations to cyclic branched covers of a knot. Limitations of the techniques will also be discussed. This is joint work with Steve Boyer and Cameron Gordon.

## 9 replies on “Investigating the L-space conjecture: Via orderability”

[…] … via orderability […]

I’m Ty, a postdoc at L’Université du Québec à Montréal, working with Steve Boyer. My research interests include mapping class groups (both big and small), and more recently, orderable groups (mostly circular).

Jonathan_Johnsonsays:

Hello Ty,

If you change the definition of CO to be invariant under both left and right multiplication, would it be possible to get a bi-orderable analog of your and Adam’s theorem?

Jonathan

Hi Jon,

It can be shown that in the bi-circularly-orderable case, the only obstruction to being biorderable is in fact having torsion (This is Proposition 3.2 in “Promoting circular orderability to left orderability”, the paper with Jason).

The proposition states that a group is bi-orderable if and only if it’s bi-CO and torsion-free.

With this in mind, you can then go on to show the following: Let G by bi-CO. Then $G \times \mathbb Z/n\mathbb Z$ is circularly orderable if and only if the only torsion elements in $G$ have order coprime to $n$.

A result like this, not precisely stated like this, will appear in the new paper (hopefully out soon!).

YingHusays:

Hi everyone,

I’m Ying. I am currently an assistant professor at the University of Nebraska Omaha.

I study foliations, flows, and orderable groups as well as other related topics in low-dimensional topology. I’ll be happy to answer any questions about my talk in the comment or in the Zoom session on Wed.

Jonathan_Johnsonsays:

Hello Ying,

Is there a characterization of when fibered knots are right-veering?

Jonathan

YingHusays:

If you know the definition of Fractional Dehn twist coefficient, then the monodromy being right-veering is equivalent to its fractional dehn twist coefficient is > 0, which is also equivalent to the degeneracy slope (which is the reciprocal of the fractional dehn twist coefficien) is positive (with the knot meridian).

A good reference for this is a paper by Kazez and Roberts, called “Fractional dehn twist coefficient in knot theory”.

I hope this helps.

HannahTurnersays:

For Xinghua: Maybe there is no way to reasonably quantify this, but – when restricting a $\widetilde{PSL(2,R)}$ representation on M to the boundary, is the hyperbolic case generic?

ChiCheukTsangsays:

Hi Xinghua,

Thank you for the talks, they are really nice!

I’m curious about how computable the translation extension locus and holonomy extension locus are. Are there ways to compute them aside from knowing $pi_1$ and working them out from there explicitly?

Also I’m a bit confused about the reasoning you mentioned during slides 5 and 6 in your second talk. Wouldn’t one have to check that the points aren’t parabolic and ideal before applying the theorems? Is this something that can be read off from the loci directly?

Thanks!