University of California, Santa Barbara Topology Seminar Fall 2016 - Spring 2017

The topology seminar is co-organized by myself (Catherine Pfaff) and Priyam Patel. Due to a scheduling conflict, Priyam was the primary organizer during the Winter 2017 quarter, but I have returned to an active organizational role.

Topology seminar Fall 2017-Spring 2018

Directions and maps for UCSB and the Mathematics Department.

KITP's guide to local accommodation

Time: Tu 3:30-4:30 (except as noted)

Place: South Hall 6635 (except as noted)



10/4/2016, Mark Hagen (Cambridge)

Title: Acylindrical hyperbolicity from cubical small-cancellation

Abstract: I will define cubical presentations and cubical small-cancellation groups, in the sense of Wise, and explain the increased flexibility this offers compared to classical small-cancellation theory. Then I'll talk about associating hyperbolic graphs to CAT(0) cube complexes, and how to use this to define a hyperbolic space on which a cubical small-cancellation group acts "nicely". From this, one can show that cubical small-cancellation groups are acylindrically hyperbolic, which has various consequences. This result generalizes the fact that strong classical small-cancellation implies hyperbolicity and is closely related to a recent result of Gruber-Sisto on acylindrical hyperbolicity of "graphical" small-cancellation groups. This talk is mainly about recent joint work with Goulnara Arzhantseva.

10/11/2016, 16:15, V4-116: Ilya Kapovich (UIUC)

Title: Cannon-Thurston maps for hyperbolic free group extensions

Abstract: Let FN be a free group of finite rank N ≥ 3 and let Γ ≤ Out(FN) be a finitely generated "convex cocompact" subgroup, that is, such that the orbit map from Γ to the free factor complex of FN is a quasi-isometric embedding. Assume also that Γ is purely atoroidal. In this case Γ determines an extension group EΓ of Γ with the quotient EΓ/FN=Γ, and it is known by a result of Dowdall and Taylor that the group EΓ is then word-hyperbolic. By a general result of Mitra the inclusion of FN in EΓ extends to a continuous surjective FN-equivariant map between their hyperbolic boundaries j: ∂ FN → ∂ EΓ, called the Cannon-Thurston map. We analyze the structure of this map and prove that the map is finite-to-one, with multiplicity at most 2N. The talk is based on a joint paper with Spencer Dowdall and Sam Taylor.

10/18/2016, Nancy Scherich (UCSB)

Title: Mapping the Braid Groups into Lattices

Abstract: Many well loved representations of the braid groups have the property that the image matrices fix a Hermitian form. Using this form, one can show that specializing to certain Salem numbers places the image inside a lattice. While it is unknown whether the image itself is a lattice, there are some interesting results on commensurability of the target lattices.


11/1/2016, Jing Tao (University of Oklahoma)

Title: Effective quasimorphisms on right-angled Artin groups

Abstract: I will discuss the construction of a large family of homogeneous quasimorphisms on right-angled Artin groups, which give rise to a lower bound of 1/24 for the stable commutator length in these groups. This work is joint with Talia Fernos and Max Forester.

11/8/2016, Mark Bell (UIUC)

Title: Nielsen--Thurston classification via the curve graph

Abstract: We will discuss a new polynomial-time algorithm, joint with Richard Webb, for computing the Nielsen--Thurston type of a mapping class. The procedure works by considering the maps action on the curve graph, which records the pairs of essential closed curves that are disjoint.

To be able to compute this action, we need to be able to construct geodesics through the curve graph. However, this graph is locally infinite and so standard pathfinding algorithms struggle. We will discuss a new refinement of the techniques of Leasure, Shackleton, Watanabe and Webb for overcoming this local infiniteness that allows such geodesics to be found in polynomial time (in terms of their length).

11/15/2016, Spencer Dowdall (Vanderbilt)

Title: Hyperbolicity in Outer space with applications to free group extensions

Abstract: The "Outer space" of the rank n free group F_n is a contractible metric space on which the Outer automorphism group Out(F_n) acts properly discontinuously. It was introduced by Culler and Vogtmann in 1986 and is now an important tool for the topological and geometric study of Out(F_n). This talk will focus on the geometry of Outer space and implications for free group extensions. The first aspects of hyperbolicity in Outer space were discovered by Algom-Kfir, who showed that axes of fully irreducible automorphisms are strongly contracting. In this talk I will present a characterization of this strongly contracting property in terms of the geodesic's projection to the free factor complex. This characterization allows one to exploit the hyperbolicity of Outer space to study many geometric aspects of free group extensions. Results here include a flexible means of producing hyperbolic free group extensions, qualitative statements regarding their structure and quasiconvex subgroups, and quantitative results about their Cannon-Thurston maps. Mostly joint with Sam Taylor, and some joint with Ilya Kapovich and Sam Taylor.

11/22/2016, Robert Kropholler (Tufts)

Title: Finiteness properties and free Abelian groups

Abstract: Finiteness properties of groups come in many flavours, I will discuss topological finiteness properties. These relate to the finiteness of skelata in a classifying space. Groups with interesting finiteness properties have been found in many ways, however all such examples contains free Abelian subgroups of high rank. I will discuss some constructions of groups discussing the various ways we can reduce the rank of free Abelian subgroups. I will introduce all the concepts required for my talk.

11/29/2016, Valentina Disarlo (Heidelberg)

Title: On the geometry of the flip graph

Abstract: The flip graph of an orientable punctured surface is the graph whose vertices are the ideal triangulations of the surface (up to isotopy) and whose edges correspond to flips. Its combinatorics is crucial in works of Thurston and Penner’s decorated Teichmuller theory. In this talk we will explore some geometric properties of this graph, in particular we will see that it provides a coarse model of the mapping class group in which the mapping class groups of some subsurfaces are convex. We will also establish upper and lower bounds on the growth of the diameter of the flip graph modulo the mapping class group, extending a result of Sleator-Tarjan-Thurston. This is a joint work with Hugo Parlier.

12/2/2016 (FRIDAY), 2pm SH6637, Jenya Sapir (UIUC)

Title: A Birman-Series result for infinitely self-intersecting geodesics on surfaces

Abstract: Let S be a hyperbolic surface. Birman and Series showed that there is a single nowhere dense, Hausdorff dimension 1 subset of S that contains the images of all complete (bi-infinite) simple geodesics. We examine sets of geodesics with different possible self-intersection rates and show that the Birman-Series type result holds for all of the sets where the rate is o(l2) intersections per length l subarc. In contrast, the set of geodesics with the maximal possible self-intersection rate, which is k l2, is dense and has full Hausdorff dimension in S.

12/6/2016, 2pm, Pallavi Dani (Louisiana State University)

Title: Commensurability classification for certain right-angled Coxeter groups

Abstract: Two finitely generated groups are abstractly commensurable if they have isomorphic finite-index subgroups. Classifying groups up to commensurability is a fundamental question in geometric group theory. I will discuss joint work with Emily Stark and Anne Thomas on the commensurability classification of a class of hyperbolic right-angled Coxeter groups. A key step in our proof is showing that these groups are virtually geometric amalgams of free groups.


1/10/2017, David Futer (Temple University)

Title: Abundant quasifuchsian surfaces in cusped hyperbolic 3-manifolds

Abstract: I will discuss a proof that every finite volume hyperbolic 3-manifold M contains an abundant collection of immersed, π1-injective surfaces. These surfaces are abundant in the sense that their lifts to the universal cover separate any pair of disjoint geodesic planes. The proof relies in a major way on the corresponding theorem of Kahn and Markovic for closed 3-manifolds. As a corollary, we recover Wise's theorem that the fundamental group of M is acts properly and cocompactly on a cube complex. This is joint work with Daryl Cooper.

1/17/2017, No Seminar

1/24/2017, Burak Ozbagci (UCLA)

Title: Fillings of unit cotangent bundles of nonorientable surfaces

Abstract: We prove that any minimal weak symplectic filling of the canonical contact structure on the unit cotangent bundle of a nonorientable closed surface other than the real projective plane is s-cobordant rel boundary to the disk cotangent bundle of the surface. If the nonorientable surface is the Klein bottle, then we show that the minimal weak symplectic filling is unique up to homeomorphism. (This is a joint work with Youlin Li.)

1/31/2017, No Seminar

2/7/2017, 5pm, Joseph Maher (CUNY College of Staten Island)

Title: Random walks on groups with coarse negative curvature.

Abstract: We will give a gentle introduction to random walks on groups satisfying various types of negative curvature conditions. A simple example is the nearest neighbour random walk on the 4-valent tree, also known as the Cayley graph of the free group on two generators. A typical random walk moves away from the origin at linear speed, and converges to one of the ends of the tree. We will discuss how to generalize this result to more general settings, such as hyperbolic groups, or acylindrical groups. This is joint work with Giulio Tiozzo.

2/14/2017, No Seminar

2/21/2017, Priyam Patel (UCSB)

Title: Algebraic and topological properties of big mapping class groups

Abstract: The mapping class group of a surface is the group of homeomorphisms of the surface up to isotopy (a natural equivalence). Mapping class groups of finite type surfaces have been extensively studied and are, for the most part, well-understood. There has been a recent surge in studying surfaces of infinite type and in this talk, we shift our focus to their mapping class groups, often called big mapping class groups. In contrast to the finite type case, there are many open questions regarding the basic algebraic and topological properties of big mapping class groups. Until now, for instance, it was unknown whether or not these groups are residually finite. We will discuss the answer to this and several other open questions after providing the necessary background on surfaces of infinite type. This work is joint with Nicholas G. Vlamis.

2/28/2017, Sergio Fenley (Florida State University)

Title: Free Seifert fibered pieces of pseudo-Anosov flows

Abstract: We prove a structure theorem for pseudo-Anosov flows restricted to Seifert fibered pieces of three manifolds. The piece is called periodic if there is a Seifert fibration so that a regular fiber is freely homotopic, up to powers, to a closed orbit of the flow. A non periodic Seifert fibered piece is called free. In this talk we consider free Seifert pieces. We show that, in a carefully defined neighborhood of the free piece, the pseudo-Anosov flow is orbitally equivalent to a hyperbolic blow up of a geodesic flow piece. A geodesic flow piece is a finite cover of the geodesic flow on a compact hyperbolic surface, usually with boundary. We introduce almost k-convergence groups, and an associated convergence theorem. We also introduce an alternative model for the geodesic flow of a hyperbolic surface that is suitable to prove these results, and we define what is a hyperbolic blow up. This is joint work with Thierry Barbot.

3/7/2017, No Seminar

3/14/2017, Stephen Trettel (UCSB)

Title: The Heisenberg Plane

Abstract: For my advancement I will talk about the Klein geometry which arises from the projective action of the Heisenberg group on the plane. This geometry is interesting in relation to other two dimensional geometries as it is the only (nontrivial) common limit of the three constant curvature geometries inside of projective geometry. However, it is also interesting in and of itself as there is a rich deformation space of Heisenberg structures on tori. I will not assume previous experience with taking limits of geometries or deformation spaces of structures, and the talk will contain a few fun mathematical animations so I hope to see you there!


4/4/2017, Nathan Williams (UCSB)

Title: Why the Fuss?

Abstract: Spherical Artin groups (like braid groups) have two surprisingly different--but equally natural--presentations. Briefly, their classical presentation arises from the chamber geometry their associated Coxeter group enjoys as a real reflection group, while their more recent dual presentation comes from interpreting the Coxeter group as a complex reflection group. In this dual presentation, certain elements of the Artin group are singled out, and these elements lead to a beautiful enumerative theory. In this talk, I will present work with C. Stump and H. Thomas that identifies a corresponding theory in the classical presentation.

4/11/2017, Martin Scharlemann (UCSB)

Title: Uniqueness in Waldhausen’s theorem (joint with M. Freedman)

Abstract: It is a famous theorem of Waldhausen that any genus g Heegaard splitting surface H in S3 is isotopic to the standard genus g splitting surface H0. Rieck’s more modern account exploits a theorem of Casson and Gordon on weakly reducible Heegaard splittings: a weak reduction of the splitting leads to a way of breaking up the surface into a connected sum of lower genus splittings, at which point induction can be used. But how unique is the isotopy of H to H0? How much is determend by the choice of a weak reduction? We discuss this in the context of a conjecture of Powell about isotopies of H0 to itself, still a work in progress.

4/18/2017, Catherine Pfaff (UCSB)

Title: Geodesics in Outer Space

Abstract: Outer automorphism groups of free groups are studied via their action on Culler-Vogtmann Outer Space. We study in particular the interplay between outer automorphism singularity invariants and the behavior of geodesics in Outer space. This is joint work with Yael Algom-Kfir, Ilya Kapovich, and Lee Mosher.

4/25/2017, Robert Edwards (UCLA), IN 4607B

Title: Embeddings of S2 into S2 x S2

Abstract: I’ll present a new theorem of Dave Gabai, which he calls the 4D Light Bulb Theorem: Everything is smooth. Suppose Σ ⊆ S2 x S2 is an embedded 2-sphere which intersects some S2 x {point} in a single point, transversally. Then Σ is unknotted, meaning that Σ can be diffeotoped to one of countably infinitely many canonical positions, indexed by the degree of the projection of Σ to the first factor S2.

This is to be compared to the classical 3D Light Bulb Theorem, where S2 x S2 is replaced by S2 x S1 and Σ is a 1-sphere, and where the conclusion is that Σ can be diffeotoped to {point} x S1.

Gabai's proof is delightfully ingenious, but not fully explainable in an hour, so this talk will consist of two parts. The first hour will outline the entire proof, presenting about 80% of its details. Then, after a break, the informal second part (30-45 minutes) will present the critical missing piece.

5/2/2017, Giuseppe Martone (University of Southern California)

Title: Positively ratioed representations and their properties

Abstract: For a hyperbolic metric on a surface, a classical construction of Bonahon describes the length of closed geodesics in terms of their geometric intersection number with a certain geodesic current. We generalize this picture to higher rank Lie groups. Namely, we associate geodesic currents to Anosov representations that satisfy an additional positivity property. We use this tool to relate the entropy and the systole length of such a positively ratioed representation. This is joint work with Tengren Zhang.

5/9/2017, Rosalie Carlson (UCSB)

Title: Finding strictly convex projective structures on manifolds

Abstract: Given a 3-manifold, can a finite time algorithm decide whether or not it has a strictly convex projective structure? Jason Manning (2002) showed such an algorithm exists for hyperbolic structures by considering representations of the fundamental group into SL(2,C). We discuss new problems that arise in the projective context, including determining whether or not a given projective structure is strictly convex, and tools such as the tautological line bundle that can be applied.

5/16/2017, Tengren Zhang (Caltech)

Title: The Goldman symplectic form and the Hitchin component.

Abstract: Let S be a closed, orientable, connected surface of genus at least 2. We prove that any ideal triangulation on S determines a symplectic trivialization (with respect to the Goldman symplectic form) of the tangent bundle of the PSL(V) Hitchin component. One can then consider the parallel flows with respect to the flat structure given by this trivialization. We give a geometric description of all such flows in terms of explicit deformations of Frenet curves, and prove that all such flows are Hamiltonian. Applying this to a particular ideal triangulation allows us to find a maximal family of Poisson commuting Hamiltonian flows on the PSL(V) Hitchin component. This generalizes the well-known fact that on Teichmuller space, the twist flows along a pants decomposition of S is a maximal family of Poisson commuting Hamiltonian flows. This is joint work with Zhe Sun and Anna Wienhard.

5/23/2017, Jenny Wilson (Stanford University)

Title: Stability in the second homology of Torelli groups

Abstract: In this talk, I will describe stability results for two families of groups, the Torelli groups of automorphisms of free groups, and the Torelli groups of mapping class groups of surfaces with one boundary component. Specifically, I will explain the following statement: the degree-2 integer homology groups of these Torelli groups are centrally stable when viewed as representations of GL_n(Z) or (respectively) Sp_{2n}(Z). This project uses a framework developed by Putnam, Church-Ellenberg-Farb, and Putnam-Sam. It is joint work with Jeremy Miller and Peter Patzt.

5/30/2017, Ryan Blair (Cal State Long Beach)

Title: Distortion and the bridge distance of knots

Abstract: We extend techniques due to Pardon to show that there is a lower bound on the distortion of a knot in R3 proportional to the minimum of the bridge distance and the bridge number of the knot. We also exhibit an infinite family of knots for which the minimum of the bridge distance and the bridge number is unbounded and Pardon's lower bound is constant.

6/6/2017, Erica Flapan (Pomona College)

Title: Intrinsic chirality of graphs in 3-manifolds

Abstract: We say that an embedding Γ of a graph γ in a 3-manifold M is achiral if there is an orientation reversing homeomorphism of (M,Γ). If a graph γ has no such embedding in M we say γ is intrinsically chiral in M.

We prove that for any closed, connected, orientable, irreducible 3-manifold M, there is an integer nM such that every 3-connected graph γ with genus(γ)> nM and no involution is intrinsically chiral in M. On the other hand, we prove that every graph has achiral embeddings in infinitely many closed, connected, orientable, irreducible 3-manifolds.