The seminar this semester will be meeting online using Zoom. The Zoom doors will open 10 minutes before the seminar at 2:50pm (EST) for chatter and self provided coffee, tea, beer, wine, cocktail, aperitif, digestif, cheese, and snacks depending on your time zone and predilection. If you would like to be put on the email list contact Ara Basmajian (abasmajian@gc.cuny.edu). The zoom link for the upcoming seminar will be included in the weekly mailing to the email list.
Time: Tuesdays 3:00pm - 4:00pm (Talks are 50 minutes long)
Ara Basmajian (CUNY, Graduate Center and Hunter College)
Email: abasmajian@gc.cuny.edu
Nick Vlamis (CUNY, Queens College)
Email: nicholas.vlamis@qc.cuny.edu
Title: Quasisymmetric Koebe Uniformization via Transboundary Modulus
Abstract: An open and connected subset of the plane is called a circle domain if all its complementary components are points or round disks. Given a metric space, X, which is homeomorphic to a circle domain whose complement has countably many components (and satisfies some additional mild conditions), we obtain a characterization of when X is quasisymmetric to a circle domain. This characterization is in terms of Schramm’s transboundary modulus and Heinonen-Koskela’s Loewner property and is inspired by Bonk’s celebrated uniformization result for planar carpets. This is joint work with Jonathan Rehmert.
Title: Homeomorphic subsurfaces and omnipresent arcs
Abstract: Essential simple arcs are ubiquitous in the study of surfaces. Up to homeomorphism, there are only finitely many types of arc on any given finite-type surface. For a surface of infinite-type, the picture is more complicated. This talk will shine a light on parts of this picture; we observe which essential arcs are more essential than others. We then arrive at, and study, a natural arc graph for infinite-type surfaces, and a well behaved (very large) subset of surfaces.
Title: Curvature bounds on least area fibers in hyperbolic 3-manifolds
Abstract: In her seminal work, Uhlenbeck investigated minimal immersions of closed surfaces into hyperbolic 3-manifolds with principal curvatures bounded by one. In the associated covering space, an almost-Fuchsian manifold, such a minimal immersion lifts to an incompressible least area embedding, and no other closed minimal surface can be found.
In contrast, the covering space associated to the fibers of closed manifolds fibering over the circle contain infinitely many homotopic least area embeddings. In joint work with Franco Vargas-Pallete, we give a “hands on” construction of infinite families of fibered hyperbolic 3-manifolds with least area minimal fibers with principal curvatures bounded below by a number strictly larger than one, and which therefore cannot be limits of almost-Fuchsian manifolds. This gives some evidence for the intuition that the geometry of the cover corresponding to a fiber should be “quite far” from that of any almost-Fuchsian manifold. This intuition was recently formalized by Huang and Lowe using different methods.
Title: Essential shifts and the asymptotic dimension of big mapping class groups.
Abstract: The Mapping class group of surfaces of infinite type are commonly referred to as big mapping class groups. Even though big mapping class groups are not countably generated, certain big mapping class groups can be generated by a coarsely bounded set and have a well defined quasi-isometry type. We study the asymptotic dimension of these groups. One feature of big mapping class groups that is not present in surfaces of finite type is the existence of shift maps, namely homomorphism that shift a subsurface into infinity. We call s shift map essential if the group generated by it is not coarsely bounded. We give a topological characterization of shifts maps that are essential and show that the presence of an essential shift map implies that the big mapping class group has an infinite asymptotic dimension. This is in contrast with the mapping class groups of surfaces of finite type where the asymptotic dimension is always finite. Joint work with Curtis Grant and Yvon Verberne.
Title: Affine transformations and isometries of infinite type flat surfaces
Abstract: We consider, for a fixed (infinite-type) surface S, the problem of determining all possible groups of isometries and Veech groups (i.e. linear parts of affine transformations) that can arise from flat metrics on S. Using ideas of Aougab, Patel and Vlamis, and of Morales and Delecroix, we show that there are topological restrictions for isometry groups, and on the other side we show evidence for the lack of obstructions for Veech groups. This is joint work in progress with Artigiani, Randecker, Sadanand & Weitze-Schmithuesen.
Title: Saddle connection complex: coarse and fine
Abstract: Translation surfaces are topological surfaces which are equipped with an Abelian differential or, in a different language, with a translation structure.
A new approach to study them is to encode their geometry in a combinatorial object, called the saddle connection complex. For translation surfaces, this complex plays the same role as its more established cousin, the arc complex, does for topological surfaces.
In the talk, I will introduce the saddle connection complex and some properties of its fine geometry (in particular Ivanov-type rigidity) and its coarse geometry (in particular no quasi-isometric rigidity).
Both is based on joint works with Valentina Disarlo, Huiping Pan, and Robert Tang.
Title: Finding and combining indicable subgroups of big mapping class groups
Abstract: In this talk, I’ll present a new way of constructing subgroups of infinite-type mapping class groups via a combination theorem for indicable subgroups. One of the applications of our main theorem is a construction for right-angled Artin groups in big mapping class, which I’ll also describe. This work is joint with Carolyn Abbott, Hannah Hoganson, Marissa Loving, and Rachel Skipper.
Title: Limits of Cubic Differentials and Real Buildings
Abstract: Consider a Riemann surface S of genus g at least 2 equipped with a holomorphic cubic differential q. For a given g, such pairs (S,q) can be used to parametrize the Hitchin component of the space of representations of the fundamental group into SL(3,R). This generalizes and extends the space of Fuchsian representations into PSL(2,R) (Teichmuller space), which corresponds to the locus q=0. This parametrization of the Hitchin component is not explicit but involves the Hitchin system of PDEs. For nonzero q, we study the asymptotic behaviour of the solution in the case of sq, as s approaches infinity, at every point on S, including the zeros of the cubic differential q. In particular, we show the geometry in this limit can be read off explicitly from q, in terms of an embedding of the universal cover of S into the real building given by the asymptotic cone of the symmetric space SL(3,R)/SO(3).