Videos of recorded lectures will be available on the Warwick Symposium YouTube channel
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Titles and abstracts
Javier Aramayona (Madrid)
The first integral cohomology of pure mapping class groups
Abstract: We will start by recalling the proof of a classical result of Powell, which asserts that mapping class groups of finite-type surfaces of genus at least three have trivial abelianization. In stark contrast, we will show that this is no longer the case if the surface is allowed to have infinite type; more concretely, we will explain how to construct non-trivial integer-valued homomorphisms from mapping class groups of infinite-genus surfaces. Further, we will give a description the first integral cohomology group of pure mapping class groups in terms of the first homology of the underlying surface. This is joint work with Priyam Patel and Nick Vlamis.
Jason Behrstock (CUNY)
Quasi-flats in hierarchically hyperbolic spaces
Abstract: Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmüller space, and others. In this talk I'll provide an introduction to studying groups and spaces from this point of view. This discussion will center around recent work in which we classify quasiflats in these spaces, thereby resolving a number of well-known questions and conjectures. This is joint work with Mark Hagen and Alessandro Sisto.
Joan Birman (Columbia)
Matt Clay (Arkansas)
Thermodynamic metrics on outer space
Abstract: I will discuss a Riemannian metric on the Culler-Vogtmann outer space arising from the tools of thermodynamic formalism. In the context of surfaces, McMullen showed that this metric is proportional to the Weil-Petersson metric on Teichmüller space. This is joint work with Tarik Aougab and Yo’av Rieck.
Valentina Disarlo (Heidelberg)
Generalized stretch lines for surfaces with boundary
Abstract: We will discuss some natural generalizations of Thurston's distance for surfaces with boundary, in particular the arc metric. We will construct a large family of geodesics for the Teichmüller space of surfaces with boundary with respect to the arc metric, which we will call "generalized stretch lines". We will prove that the Teichmüller space with the arc metric is a geodesic metric space, and that it is a Finsler space. This generalizes a result by Thurston on punctured surfaces. This is joint work with Daniele Alessandrini (University of Heidelberg).
Spencer Dowdall (Vanderbilt)
Abstract commensurations of big mapping class groups
Abstract: It is a classic result of Ivanov that the mapping class group of a finite-type surface is equal to its own automorphism group. Relatedly, it is well-known that non-homeomorphic surfaces cannot have isomorphic mapping class groups. In the setting of "big mapping class groups" of infinite-type surfaces, the situation is more complicated due to the fact that the sheer enormity and variety of behavior prevents group elements from having canonical descriptions in terms of normal forms. This talk will present work with Juliette Bavard and Kasra Rafi overcoming these difficulties and extending the above results to big mapping class groups. In particular, we show that any isomorphism between big mapping class groups is induced by a homeomorphism of the surfaces and that each big mapping class group is equal to its abstract commensurator.
Matthew Durham (Yale)
Geometrical finiteness and Veech subgroups of mapping class groups
Abstract: I will discuss work in progress with Dowdall, Leininger, and Sisto, in which we aim to develop a notion of geometrical finiteness for subgroups of mapping class groups. Motivated by the theory of convex cocompact subgroups, which are precisely those which determine hyperbolic surface group extensions, I will describe some hyperbolic properties of the surface group extensions coming from lattice Veech subgroups.
Viveka Erlandsson (Bristol)
Counting curves in orbifolds
Abstract: Mirzakhani proved the asymptotic growth of the number of curves on a surface of bounded hyperbolic length, in each mapping class group orbit, as their length grows. Lately I have been giving talks explaining how to extend this result to other notions of lengths of curves. In this talk I will go in a different direction and explain why Mirzakhani’s original result holds when the surface is replaced with an orbifold. This is joint work with Juan Souto.
Vaibhav Gadre (Glasgow)
Cusp excursions of random geodesics for Weil--Petersson type metrics
Abstract: In this talk, we will consider cusp excursions of random geodesics for Weil--Petersson type metrics on orientable surfaces of finite type. In particular, we will explain how to derive bounds for the size of the "maximal" excursions. Furthermore, assuming polynomial mixing for the Weil--Petersson flow on non-exceptional moduli spaces, we will give similar bounds in the general setting. Our methods rely on deriving effective ergodic average for mixing flows. This is joint work with Carlos Matheus.
Autumn Kent (UW Madison)
On word hyperbolic surface bundles.
Abstract: There is a characterization of hyperbolicity of the fundamental group of a surface bundle due to Farb-Mosher-Hamenstaedt, namely that the bundle has hyperbolic fundamental group if and only if the fundamental group is ``convex cocompact,'' a notion analogous to the synonymous notion in Kleinian groups. I will discuss joint work with Bestvina, Bromberg, and Leininger that gives a new characterization of convex cocompactness, namely that the group is purely pseudo-Anosov and undistorted in the mapping class group.
Dan Margalit (GA Tech)
Mapping class groups in complex dynamics
Abstract: A polynomial can be thought of as a branched cover of the Riemann sphere. Given a branched cover of the sphere, we can ask if it is equivalent to a polynomial, and if so, which one. In 2006, Bartholdi and Nekrashevych gave an algebraic solution to this problem and applied their method to explicitly solve the twisted rabbit problem of Douady and Hubbard. With Belk, Lanier, and Winarski, we give a purely topological method that closely mirrors the modern theories of mapping class groups and outer automorphisms of free groups.
Howard Masur (U Chicago)
Teichmüller geodesics that diverge on average in moduli space
Abstract: Given a flow on a topological space, a trajectory is said to diverge on average if for any compact subset K the average amount of time the trajectory spends in K goes to zero. I will talk about the meaning of this is the context of geodesics on the modular curve and then discuss results about divergent on average Teichmüller geodesics in moduli space in higher genus. This is joint work with Paul Apisa.
Yair Minsky (Yale)
Weil-Petersson geometry, Dehn filling and branched surfaces
Abstract: We study pseudo-Anosov mapping classes with bounded normalized Weil-Petersson translation distance (and unbounded genus). In analogy with a result of Farb-Leininger-Margalit for Teichmüller translation distances, we show all such mapping classes fit together into a finite collection of cusped hyperbolic 3-manifolds, where the cusps are filled to become either vertical (transverse to fibers) or horizontal (parallel to fibers). After a reduction using work of Schlenker, Kojima-McShane and Brock-Bromberg, the argument uses the theory of branched surfaces in 3-manifolds. This is joint work with Chris Leininger, Juan Souto and Sam Taylor.
Priyam Patel (UCSB)
Lifting curves on surfaces via 3-manifolds and the curve complex.
Abstract: Given two simple closed curves alpha and beta intersecting many times on an orientable surface S, we are interested in studying the minimal degree of a finite cover of S such that there is a lift of alpha disjoint from a lift of beta. In joint work with Tarik Aougab and Sam Taylor, we use the geometry of hyperbolic 3-manifolds to obtain lower bounds on this degree in terms of curve complex distance between the curves alpha and beta. After describing some of the techniques we use, I will highlight an interesting application of our work that gives lower bounds on the degrees of special covers for certain cube complexes associated to surfaces.
Andrew Putman (Notre Dame)
The Johnson filtration is finitely generated
Abstract: A recent breakthrough of Ershov-He shows that the Johnson kernel subgroup of the mapping class group is finitely generated for g at least 12. In joint work with Ershov and Church, I have extended this to show that every term of the lower central series of the Torelli group is finitely generated once the genus is sufficiently large. A byproduct of our work is a proof that the Johnson kernel is finitely generated for g at least 4 which is remarkably simple (so simple, in fact, that I will be able to give in nearly complete detail in this talk).
Kasra Rafi (Toronto)
Strong Contractibility of geodesics in the mapping class group
Abstract: We show that the axis of a pseudo-Anosov homeomorphism in the mapping class group may not have the strong contractibility property. Specifically, we show that, after choosing a generating set carefully, one can find a pseudo-Anosov homeomorphism f, a sequence of points x_k and a sequence of radii R_k so that the ball B(x_k, R_k) is disjoint from the axis of f, but the closest point projection of B(x_k, R_k) to the axis of f has a diameter at least c log(R_k). Along the way we show how, in certain situations, the distance between two points in the mapping class group can be computed precisely, not just up to a multiplicative error. This is a joint work in Yvon Verberne.
Jenya Sapir (Binghamtom)
Tessellations from long geodesics on surfaces
Abstract: I will talk about a recent result of Athreya, Lalley, Wroten and myself. Given a hyperbolic surface S, a typical long geodesic arc will divide the surface into many polygons. We give statistics for the geometry of a typical tessellation. Along the way, we look at how very long geodesic arcs behave in very small balls on the surface.
Balazs Strenner (GA Tech)
Fast computation in mapping class groups
Abstract: I will give an update on our project with Dan Margalit and Oyku Yurttas whose goal is to give a framework for fast computation in mapping class groups. We show that there is a quadratic-time algorithm that computes the Nielsen-Thurston type of a mapping class (finite order, pseudo-Anosov or reducible). The algorithm also finds the reducing curves and the stretch factors and invariant foliations on pseudo-Anosov components. An implementation is now under way as well.
Jing Tao (Oklahoma)
Fine geometry of the Thurston metric
Abstract: The Thurston metric is an asymmetric metric on Teichmuller Space defined using Lipschitz constants of maps between hyperbolic surfaces. This metric was introduced by Thurston in the late 80's, who showed this metric is geodesic, though geodesics are not necessarily unique, and induced by an asymmetric Finsler norm on tangent space. In this talk, I will survey some results about the fine geometry in the case of the punctured torus. This talk is based on joint work with David Dumas, Anna Lenzhen, and Kasra Rafi.
Sam Taylor (Temple)
Random veering triangulations are not geometric
Abstract: Veering triangulations were introduced by Agol as canonical ideal triangulations of pseudo-Anosov mapping tori which are punctured along singular orbits. Since these manifolds are hyperbolic, Agol asked whether veering triangulations are always geometric. That is, can the triangulation be realized in the hyperbolic metric with each tetrahedron positively oriented? Unfortunately, the answer is no since Hodgson, Issa, and Segerman subsequently produced examples of nongeometric veering triangulations. In this talk, I’ll discuss joint work with Futer and Worden that explains why this is generally the case.
Richard Webb (Cambridge)
Algorithms for surfaces and their mapping class groups
Abstract: I will survey state-of-the-art joint work with Mark Bell on various polynomial-time algorithms. The joint work covers topological problems such as finding the minimal position of multiarcs/curves on a surface, geometric problems such as finding geodesics in the curve graph, and algebraic problems such as solving the conjugacy problem for the mapping class group. I will discuss the complexities of these algorithms and then pose some curious questions.
Oyku Yurttas (Dicle)
Algorithms for multicurves with Dynnikov coordinates
Abstract: We use Cumplido’s relaxation algorithm for multicurves, and calculate in polynomial time the number of connected components of a multi curve, and the geometric intersection number of two multicurves on the n-punctured disk, taking as input their Dynnikov coordinates. This is joint work with Toby Hall.