Schedule for: 19w5044 - Ordered Groups and Rigidity in Dynamics and Topology

This talk will be an introduction to ordered groups and their connections with topology. Beginning with learning of the left-orderability of braid groups some decades ago, I will describe how I came to fall in love with the subject.

I will discuss practical computational techniques for proving the fundamental group of a closed hyperbolic 3-manifold is or is not orderable and describe the result of applying them to some 300,000 manifolds. I will include discussion of a new practical algorithm for solving the word problem for the fundamental group of a hyperbolic 3-manifold based on interval arithmetic and effective versions of the inverse function theorem.

The Dynkin diagrams of types A,D and E arise in many classification problems in mathematics. We conjecture a modest addition to this list: the fibered links that induce the standard tight contact structure on $S^3$ and have some cyclic branched cover whose fundamental group is not left-orderable. We will discuss progress towards a proof of this conjecture. This is joint work with Michel Boileau and Steve Boyer.

We formulate a ping-pong lemma for actions of fundamental groups of graph of groups, listing sufficient conditions for the action to be faithful. This allows to generalize the notion of ping-pong partitions for actions on the circle (appearing in recent works by Matsumoto and Mann-Rivas), to the case of groups with torsion. In concrete terms, a ping-pong partition is a collection of finitely many combinatorial data that determines the semi-conjugacy class of the action. We apply this to give a partial positive solution to an old conjecture by Dippolito on codimension-one foliations with exceptional minimal set. This is part of a long-term project with many collaborators.

A left-order on a group G is totally determined by its positive cone P, that is the elements in G that are greater than the identity. The set P is a subsemigroup and we can ask ourselves when this semigroup can befinitely generated, described by a regular language,etc. A recent resultof S.Hermiller and Z.Sunic shows that non-abelian free groups do not have regular positivecones. I will discuss this and how to generalize this result to limit groups. The talk will be based on joint work with C.Rivas and I will report some results of my PhD student H.L.Su. (Yago Antolin is from the Unversidad Autónoma de Madrid)

A positive cone in a finitely generated group is called regular if it can be represented by a regular language over the group-generators. In this talk, I will discuss examples and non-examples of groups supporting regular positive cones. For instance we will see that every left-orderable polycyclic group admits a regular positive cone, but this is no longer true for the class solvable groups. I will also try to relate to presence of regular positive cone with the presence/absence of isolated orders.

A notion of laminar groups was introduced by D. Calegari. A group acting on the circle by orientation-preserving homeomorphisms is called a laminar group if it admits an invariant lamination. Abundant examples arise naturally in the study of low-dimensional topology and geometric group theory. We will discuss how topology, geometry, and dynamics interplay when we study laminar groups. Some old and new results will be discussed as examples.

Let $Diff^r(S^1)$ denote the group of orientation-preserving $C^r$ diffeomorphisms on the circle. We prove that a right-angled Artin group $G$ embeds into $Diff^2(S^1)$ if and only if $G$ does not contain $(F_2 x Z) * Z$. This extends a previous joint work with Baik and Koberda, and answers a question of Kharlamov. (joint work with Koberda)

I will present a new characterisation for a circularly-orderable group to be left-orderable, and introduce the obstruction spectrum of a circularly-orderable group. This raises a plethora of intriguing questions. This is joint work with Jason Bell and Adam Clay.

If a 3-manifold group does not admit a bi-ordering, then we may expect that it has a generalized torsion element. As a particular case, the fundamental group of any 3-manifold obtained by non zero surgery on a knot in the 3-sphere may have such an element. Then there are two situations: (1) a generalized torsion element in a knot group becomes a generalized torsion element in the surgered 3-manifold, or (2) a generalized torsion element arises via the Dehn filling. The first situation leads us to study of normal closures of slope elements in a knot group. In the first talk we investigate relationships among such normal subgroups. In particular, we establish the peripheral Magnus property. In the second talk we focus on generalized torsion elements in Dehn surgered manifolds arisen from the first or the second situations. We also take a closer look at some explicit examples. In the third talk we prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable 3-manifold M has a generalized torsion element if and only if the fundamental group of some prime factor of M has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal 3-manifolds whose fundamental group has a generalized torsion element, while the fundamental group of each decomposing piece has no such elements. Additionally, in the course of the proof of the former result, we give an upper bound for the stable commutator length of generalized torsion elements.

If a 3-manifold group does not admit a bi-ordering, then we may expect that it has a generalized torsion element. As a particular case, the fundamental group of any 3-manifold obtained by non zero surgery on a knot in the 3-sphere may have such an element. Then there are two situations: (1) a generalized torsion element in a knot group becomes a generalized torsion element in the surgered 3-manifold, or (2) a generalized torsion element arises via the Dehn filling. The first situation leads us to study of normal closures of slope elements in a knot group. In the first talk we investigate relationships among such normal subgroups. In particular, we establish the peripheral Magnus property. In the second talk we focus on generalized torsion elements in Dehn surgered manifolds arisen from the first or the second situations. We also take a closer look at some explicit examples. In the third talk we prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable 3-manifold M has a generalized torsion element if and only if the fundamental group of some prime factor of M has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal 3-manifolds whose fundamental group has a generalized torsion element, while the fundamental group of each decomposing piece has no such elements. Additionally, in the course of the proof of the former result, we give an upper bound for the stable commutator length of generalized torsion elements.

If a 3-manifold group does not admit a bi-ordering, then we may expect that it has a generalized torsion element. As a particular case, the fundamental group of any 3-manifold obtained by non zero surgery on a knot in the 3-sphere may have such an element. Then there are two situations: (1) a generalized torsion element in a knot group becomes a generalized torsion element in the surgered 3-manifold, or (2) a generalized torsion element arises via the Dehn filling. The first situation leads us to study of normal closures of slope elements in a knot group. In the first talk we investigate relationships among such normal subgroups. In particular, we establish the peripheral Magnus property. In the second talk we focus on generalized torsion elements in Dehn surgered manifolds arisen from the first or the second situations. We also take a closer look at some explicit examples. In the third talk we prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental group of a compact orientable 3-manifold M has a generalized torsion element if and only if the fundamental group of some prime factor of M has a generalized torsion element. On the other hand, we demonstrate that there are infinitely many toroidal 3-manifolds whose fundamental group has a generalized torsion element, while the fundamental group of each decomposing piece has no such elements. Additionally, in the course of the proof of the former result, we give an upper bound for the stable commutator length of generalized torsion elements.

The L-space conjecture relates non-left-orderability of 3-manifold groups to Heegaard Floer homology lens-spaces, or, L-spaces. In this talk I will give the definition of an L-space , and attempt to give a feeling for this class of manifolds by focusing on some examples. One class of examples is due to Ozsváth and Szabó: branched double covers of the 3-sphere with branch set a non-split alternating link. This leads to a surprising conjecture (implied by the L-space conjecture) relating simplicity in Khovanov homology to non-left-orderability of the fundamental group of the branched double cover.

The L-Space Conjecture is taking the low-dimensional topology community by storm. It aims to relate seemingly distinct Floer homological, algebraic, and geometric properties of a closed 3-manifold Y. In particular, it predicts a 3-manifold Y isn't "simple" from the perspective of Heegaard-Floer homology if and only if Y admits a taut foliation. The reverse implication was proved by Ozsvath and Szabo. In this talk, we'll present a new theorem supporting the forward implication. Namely, we'll build taut foliations for manifolds obtained by surgery on positive 3-braid closures. Our theorem provides the first construction of taut foliations for every non-L-space obtained by surgery along an infinite family of hyperbolic L-space knots. As an example, we'll construct taut foliations in every non-L-space obtained by surgery along the P(-2,3,7) pretzel knot.

A branched surface that meets the torus boundary of a compact 3-manifold transversely (in a train track \tau) can sometimes be ``upgraded'' to a branched surface that fully carries CTFs that strongly realize all boundary slopes except one. We give a condition on \tau that guarantees that such an upgrade is possible. This approach succeeds for all alternating and Montesinos knots without L-space surgeries, for certain Murasugi sums, and for all nontrivial connected sums of alternating knots, Montesinos knots, or fibered knots. This is joint work with Charles Delman.

Given a compact space endowed with a flow, every group of orbit-preserving homeomorphisms of the space naturally acts on the real line (identified with an orbit of the flow). This simple observation can be used to define interesting examples of left-orderable groups. In a joint work with Michele Triestino, we explore this idea by defining and studying a class of groups acting on suspension flows of homeomorphisms of the Cantor set. I will explain how this can be used to give a short and conceptual construction of finitely generated simple left-orderable groups, whose existence was recently obtained by Hyde and Lodha. I will also discuss several additional properties of these groups, such as the inability to act on the circle without fixed points, and the lack of subgroups with property (T).

This talk is motivated by the conjecture: the fundamental group of a QHS is left-orderable if and only if it admits a co-orientable taut foliation. It is known that if the Euler class of the taut foliation vanishes, then the fundamental group is left-orderable. In this talk, we will investigate the Euler class of co-orientable taut foliations on 3-manifolds which are obtained by Dehn surgery along a null-homologous knot.

Consider a one-parameter group of homeomorphisms of some interval $I$, i.e. a $C^0$ flow on I, and assume its time-1 and $\alpha$ maps are smooth, for some irrational number $\alpha$. Does this imply that the flow itself is smooth? We will explain that the answer depends on the arithmetic nature of $\alpha$, namely: the answer is yes if alpha is diophantine, and no otherwise. For flows without fixed points, this follows from famous linearization results for circle diffeomorphisms, and we will see how some of the techniques from this context adapt to our general situation.

Let G be a group and H be a subgroup of G. We say that H is left relatively convex in G if the left G-set G/H has at least one G-invariant order. When G is left orderable, this holds if and only if H is convex in G under some left ordering of G. We give a criterion for H to be left relatively convex in G that generalizes a well known criterion of Burns and Hale. We then use this criterion to show that all maximal cyclic subgroups are left relatively convex in free groups, in right-angled Artin groups, and in surface groups that are not the Klein-bottle group. The free-group case extends a result of Duncan and Howie. We show that if G is left orderable, then each free factor of G is left relatively convex in G. More generally, for any graph of groups, if each edge group is left relatively convex in each of its vertex groups, then each vertex group is left relatively convex in the fundamental group; this generalizes a result of Chiswell. Finally, we show that all maximal cyclic subgroups in locally residually torsion-free nilpotent groups are left relatively convex. This is a joint work with Yago Antolin and Warren Dicks.

This will be a session for free discussion of open problems.