Starting in Spring 2023 the seminar will go back to in-person talks. We will also attempt to run the seminar as hybrid using zoom. Zoom doors will open a few minutes before the seminar 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.
Tuesdays 2:45pm–3:45pm, room 5417. ***Note: we are back to an afternoon time.
Organizers
Ara Basmajian (CUNY, Graduate Center and Hunter College)
Email: abasmajian@gc.cuny.edu
Dragomir Saric (CUNY, Graduate Center and Queens College)
Email: dragomir.saric@qc.cuny.edu
Nick Vlamis (CUNY, Graduate Center and Queens College)
Email: nvlamis@gc.cuny.edu
Title: Non-orientable surfaces and mapping class group orbit closures
Abstract: The space of measured laminations on a surface—which can be viewed as the closure of the set of weighted simple closed curves—is a fundamental tool when studying hyperbolic surfaces. When the surface is orientable, this space and the action of the mapping class group on it is by now well-understood. However, the situation is very different in the non-orientable case. For example, while almost all projective laminations are uniquely ergodic in the orientable case, this is far from true in the non-orientable setting: those that have a closed (one-sided) leaf in its support have full measure. Likewise, minimality of the action of the mapping class group on PML and Lindenstrauss-Mirzakhani’s orbit closure classification fail in the non-orientable setting. In this talk I will discuss these differences and some recent results. In particular, I will describe the orbit closures of the mapping class group action on the space of measured laminations, which is joint work with Gendulphe, Pasquinelli, and Souto.
Title: Circle homeomorphisms with square summable diamond shears
Abstract: In this talk, I will introduce two l^2 spaces of homeomorphisms of the circle (up to Mobius transformations) defined in terms of shear and diamond shear coordinates along the edges of the Farey tessellation. Shears and diamond shears are related combinatorial, and diamond shears are also closely related to another set of coordinates called log lambda lengths. I will motivate the definition of diamond shear coordinates by their relationship to Weil-Petersson homeomorphisms in the finite support case, and explain how maps with l^2 shears and diamond shears are related to Holder classes of circle homeomorphisms and the Weil-Petersson class. The material of this talk is joint work with Dragomir Šarić and Yilin Wang from https://arxiv.org/abs/2211.11497.
Title: Geometry of geodesic currents
Abstract: The space of projective, filling currents PFC(S) contains many structures relating to a closed, genus g surface S. For example, it contains the set of all closed curves on S, as well as an embedded copy of Teichmuller space, and many other spaces of metrics on S. We will discuss a structure theorem that compares each filling current with a suitably chosen point in Teichmuller space. We will then use this structure theorem to explore the geometry of PFC(S) under an extension of the Thurston metric.
Title: Sullivan’s Dictionary, Limits of deformations, and Modular Laminations
Abstract: Sullivan’s dictionary between Kleinian groups and rational maps reveals how many objects, such as limit sets and Julia sets, are different names for the same thing. On a deeper level, it provides conjectures in one field that are analogs of well-known theorems in the other. One such well-known theorem, proven by W. Thurston, is the compactness of the space of representations (in Isom(H^3)) of the fundamental group of a compact 3-manifold with acylindrical boundary. The analog of this theorem for rational maps was conjectured by C. McMullen in the early 1990’s. Because there is no quotient three-manifold for a rational map, new tools are needed to study degenerating sequences of deformations, so we introduce the concept of an invariant modular lamination and use it to prove this conjecture.
Title: Renormalized volume for Schottky groups
Abstract: In this talk I will discuss the renormalization of volume for convex co-compact hyperbolic handlebodies, also known as Schottky groups. I will address a question of Maldacena inspired by the relation between renormalized volume and entropy, and I'll describe how this relates to extremal lengths and determinant of the Laplacian. Part of this talk is based on upcoming joint work with Jeff Brock.
Title: Normal generators of mapping class groups
Abstract: We will discuss how to show a given mapping class is and is not a normal generator of the mapping class group, and then discuss related open and closed questions.
Title: Surface Homeomorphisms and certain Algebraic Units
Abstract: Homeomorphisms from a compact topological surface to itself have a topological invariant associated to them known as the stretch-factor or dilatation, which is a positive real number. Logarithms of this number represent lengths of closed geodesics in the Moduli space of complete hyperbolic metrics of constant curvature on the surface, as well as the topological entropy of the homeomorphism, among other things. The precise nature of what numbers appear as stretch-factors is unknown. It is known since the 80s that the numbers are algebraic integers, units in fact, whose Galois conjugates are bounded in absolute value between the number and its reciprocal. One might suppose that these properties characterize all stretch-factors, but this isn't known. In the talk, we will make a case for and against this supposition, and provide interesting results along the way.
Title: Ergodicity of the geodesic flow on symmetric surfaces
Abstract: An infinite Riemann surface is symmetric if it has an orientation-reversing isometry whose fixed points are closed and/or bi-infinite geodesics. We prove that the geodesic flow on the unit tangent bundle of a symmetric surface is ergodic if and only if the covering Fuchsian group is of the first kind.
Basmajian, Hakobyan, and Saric gave a sufficient condition on the Fenchel-Nielsen parameters for a Riemann surface to have ergodic geodesic flow. A planar Riemann surface is called a flute if it has countably many punctures that accumulate to a single topological end. A flute surface with relative twists 21 is symmetric. We use our results to com- pletely determine which half-twist flute surfaces have ergodic geodesic flow complementing the results of Basmajian, Hakobyan, and Saric. Our results extends to symmeteric surfaces with infinite genus and finitely many ends. This is a joint work with D. Saric.
Title: Coarse Geometry of Big Mapping Class Groups of Graphs
Abstract: Motivated by the recent interest in big mapping class groups of surfaces, we will introduce an analogue of big mapping class groups as defined by Algom-Kfir and Bestvina that hopes to answer the question: What is “Big Out(Fn)”? This group will consist of proper homotopy classes of proper homotopy equivalences of locally finite, infinite graphs. These groups are "huge groups" in that they are non-locally compact topological groups. We will discuss some classification theorems related to the coarse geometry of these groups using the machinery of Rosendal. This is joint work with Hannah Hoganson and Sanghoon Kwak.
Title: Divergent geodesics in the universal Teichmüller space
Abstract: Thurston boundary of the universal Teichmüller space is identified with the space of projective bounded measured laminations. A geodesic ray in the universal Teichmüller space is of generalized Teichmüller type if it shrinks the vertical foliation of a holomorphic quadratic differential. In the previous work of Hakobyan and Šarić, existence of limits at infinity was proved for many generalized Teichmüller rays in the universal Teichmüller space. In this paper we provide the first example of generalized Teichmüller rays which do not have a unique limit point in Thurston boundary and completely determine the limit sets of these rays in the space of projective bounded measured laminations. This is a joint work with Hrant Hakobyan.
Title: Towards the Margulis constant for hyperbolic 3-manifolds
Abstract: For a closed hyperbolic 3-manifold M, let M_\epsilon be the set of all points in M whose injectivity radius is less than \epsilon. A lemma of Margulis implies that there is a universal constant \mu_3 such that if \epsilon < \mu_3 then M_\epsilon is topologically a disjoint union of solid tubes. This constant is useful for many finiteness arguments in hyperbolic geometry as well as effective Dehn surgery. The true value of \mu_3 is unknown, but lower bounds exist. Meyerhoff shows that \mu_3 > 0.108, while Culler, Shalen, and others have better lower bounds when the choice of M is restricted. In this talk, we will describe partial progress towards the exact value of \mu_3. In particular, we will explain how a symmetric variant of this constant can be computed and how it can be used, in preliminary computations, to suggest that \mu_3 > 0.5. This is joint work with David Gabai and David Futer.
History of the hyperbolic geometry seminar