Explanipedia

Ordered bases, order-preserving automorphisms and bi-orderable link groups Open

Tommy Wuxing Cai, Adam Clay, Dale Rolfsen · 2025

We give a new criterion which guarantees that a free group admits a bi-ordering that is invariant under a given automorphism. As an application, we show that the fundamental group of the “magic manifold” is bi-orderable, answering a questi…

Circular-orderability of 3-manifoldgroups Open

Idrissa Bâ, Adam Clay · 2025

Order-detection and non-left-orderable surgeries on links Open

Adam Clay, Junyu Lu · 2025

Beginning with a $3$-manifold $M$ having a single torus boundary component, there are several computational techniques in the literature that use a presentation of the fundamental group of $M$ to produce infinite families of Dehn fillings …

Generalized torsion in amalgams Open

Tommaso Cai, Adam Clay · 2025

We give a condition sufficient to ensure that an amalgam of groups is generalized torsion-free. As applications, we construct a closed 3-manifold whose fundamental group is generalized torsion-free and non bi-orderable; a one-relator group…

Graph manifolds that admit arbitrarily many Anosov flows Open

Adam Clay, Tali Pinsky · 2024

For each natural number n , we construct an example of a graph manifold supporting at least n different Anosov flows that are not orbit equivalent. Our construction is reminiscent of the Thurston-Handel construction [11]: we cut a geodesic…

Condensation and left-orderable groups Open

Adam Clay · 2024

We discuss condensed left-orderings and develop new techniques to show that the conjugacy relation on the space of left-orderings is not smooth. These techniques apply to the solvable Baumslag Solitar groups and to Thompson’s group .

Ordered bases, order-preserving automorphisms and bi-orderable link groups Open

Tommy Wuxing Cai, Adam Clay, Dale Rolfsen · 2024

We give a new criterion which guarantees that a free group admits a bi-ordering that is invariant under a given automorphism. As an application, we show that the fundamental group of the "magic manifold" is bi-orderable, answering a questi…

Order-detection of slopes on the boundaries of knot manifolds Open

Steven Boyer, Adam Clay · 2024

Motivated by the L-space conjecture, we investigate various notions of order-detection of slopes on knot manifolds. These notions are designed to characterise when rational homology 3-spheres, obtained by gluing compact manifolds along tor…

Order-detection, representation-detection, and applications to cable knots Open

Adam Clay, Junyu Lu · 2024

Given a $3$-manifold $M$ with multiple incompressible torus boundary components, we develop a general definition of order-detection of tuples of slopes on the boundary components of $M$. In parallel, we arrive at a general definition of re…

Rigidity of braid group actions on $\mathbb{R}$ and of low-genus mapping class group actions on $S^1$ Open

Idrissa Bâ, Adam Clay, Tyrone Ghaswala · 2023

Every nontrivial action of the braid group $B_n$ on $\mathbb{R}$ by orientation-preserving homeomorphisms yields, up to conjugation by a homeomorphism of $\mathbb{R}$, a representation $ρ: B_n \rightarrow \mathrm{H\widetilde{ome}o}_+(S^1)$…

Condensation and left-orderable groups Open

Filippo Calderoni, Adam Clay · 2023

We discuss condensed left-orderings and develop new techniques to show that the conjugacy relation on the space of left-orderings is not smooth. These techniques apply to the solvable Baumslag Solitar groups and to Thompson's group F.

The Borel complexity of the space of left-orderings, low-dimensional topology, and dynamics Open

Filippo Calderoni, Adam Clay · 2023

We develop new tools to analyze the complexity of the conjugacy equivalence relation $E_\mathsf{lo}(G)$, whenever $G$ is a left-orderable group. Our methods are used to demonstrate non-smoothness of $E_\mathsf{lo}(G)$ for certain groups $G…

Cofinal elements and fractional Dehn twist coefficients Open

Adam Clay, Tyrone Ghaswala · 2022

We show that for a surface $S$ with positive genus and one boundary component, the mapping class of a Dehn twist along a curve parallel to the boundary is cofinal in every left ordering of the mapping class group $\operatorname{Mod}(S)$. W…

Order-detection of slopes on the boundaries of knot manifolds Open

Steven Boyer, Adam Clay · 2022

Motivated by the L-space conjecture, we investigate various notions of order-detection of slopes on knot manifolds. These notions are designed to characterise when rational homology 3-spheres obtained by gluing compact manifolds along toru…

The number of locally invariant orderings of a group Open

Idrissa Bâ, Adam Clay, Ian M. Thompson · 2022

We show that if a nontrivial group admits a locally invariant ordering, then it admits uncountably many locally invariant orderings. For the case of a left-orderable group, we provide an explicit construction of uncountable families of loc…

Borel structures on the space of left‐orderings Open

Filippo Calderoni, Adam Clay · 2022

In this paper we study the Borel structure of the space of left-orderings $\mathrm{LO}(G)$ of a group $G$ modulo the natural conjugacy action, and by using tools from descriptive set theory we find many examples of countable left-orderable…

Circular orderability of 3-manifold groups Open

Idrissa Bâ, Adam Clay · 2021

This paper initiates the study of circular orderability of $3$-manifold groups, motivated by the L-space conjecture. We show that a compact, connected, $\mathbb{P}^2$-irreducible $3$-manifold has a circularly orderable fundamental group if…

Promoting circular-orderability to left-orderability Open

Jason P. Bell, Adam Clay, Tyrone Ghaswala · 2021

Motivated by recent activity in low-dimensional topology, we provide a new criterion for left-orderability of a group under the assumption that the group is circularly-orderable: A group is left-orderable if and only if is circularly-ord…

The space of circular orderings and semiconjugacy Open

Adam Clay, Idrissa Bâ · 2020

Work of Linnell shows that the space of left-orderings of a group is either finite or uncountable, and in the case that the space is finite, the isomorphism type of the group is known---it is what is known as a Tararin group. By defining s…

Three manifolds that admit infinitely many Anosov flows Open

Adam Clay, Tali Pinsky · 2020

We construct an example of a graph manifold that supports infinitely many Anosov flows that are not orbit equivalent. Our construction is reminiscent of the Thurston-Handel construction, consisting of cutting open a geodesic flow on a surf…

Graph manifolds that admit arbitrarily many Anosov flows Open

Adam Clay, Tali Pinsky · 2020

For each natural number n, we construct an example of a graph manifold supporting at least n different Anosov flows that are not orbit equivalent. Our construction is reminiscent of the Thurston-Handel construction: we cut a geodesic flow …

Generalizations of the Burns–Hale theorem Open

Adam Clay · 2020

The Burns-Hale theorem states that a group G is left-orderable if and only if G is locally projectable onto the class of left-orderable groups. Similar results have appeared in the literature in the case of UPP groups and Conradian left-or…

Dense Orderings in the Space of Left-orderings of a Group Open

Adam Clay, Tessa Reimer · 2019

Every left-invariant ordering of a group is either discrete, meaning there is a least element greater than the identity, or dense. Corresponding to this dichotomy, the spaces of left, Conradian, and bi-orderings of a group are naturally pa…

Free products of circularly ordered groups with amalgamated subgroup Open

Adam Clay, Tyrone Ghaswala · 2019

This paper gives necessary and sufficient conditions that the free product\nwith amalgamation of circularly-ordered groups admit a circular ordering\nextending the given orderings of the factors. Our result follows from\nestablishing a cat…

On the number of circular orders on a group Open

Adam Clay, Kathryn Mann, Cristóbal Rivas · 2018

Foliations, orders, representations, L-spaces and graph manifolds Open

Steven Boyer, Adam Clay · 2017

Free groups, surface groups and covering spaces Open

Adam Clay, Dale Rolfsen · 2016

Automorphisms acting on the left-orderings of a bi-orderable group Open

Adam Clay, Sina Zabanfahm · 2016

We generalize a result of T. Koberda by showing that the natural action of the automorphism group on the space of left-orderings is faithful for all nonabelian bi-orderable groups G, as well as for a certain class of left-orderable groups …

Ordered groups and topology Open

Adam Clay, Dale Rolfsen · 2015

This is a draft of a book submitted for publication by the AMS. Its theme is the remarkable interplay, accelerating in the last few decades, between topology and the theory of orderable groups, with applications in both directions. It begi…

Slope detection, foliations in graph manifolds, and L-spaces Open

Steven Boyer, Adam Clay · 2015

A graph manifold rational homology $3$-sphere $W$ with a left-orderable fundamental group admits a co-oriented taut foliation, though it is unknown whether it admits a smooth co-oriented taut foliation. In this paper we extend the gluing t…

Adam Clay YOU? Author Swipe