The hyperbolic geometry seminar is based at the Graduate Center of the City University of New York (365 5th Ave, New York, NY 10016). All are welcome. It is run as a hybrid seminar with in-person talks. For those who want to zoom-in doors 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 one of the organizers listed below. The zoom link for an upcoming seminar is included in the weekly email list.
Tuesdays 2:45pm–3:45pm, room C415A. (C-level in the basement)
September 15: Matthew Stover (CUNY, Graduate Center)
Title: Rigidity of the moduli space of cubic surfaces
Abstract: This talk will be on joint work with Gregorio Baldi, Benson Farb, and Ariyan Javanpeykar showing that the moduli space of smooth complex cubic surfaces in CP^3 is trivial. This is analogous to classical theorems for the moduli space of Riemann surfaces of genus g, etc. The proof uses an incomplete complex hyperbolic uniformization studied by Carlson and Toledo and connections with a certain Artin group, which both open the door to techniques from geometric topology.
September 22: Jayadev Athreya (Fordham U.)
Title: Pisot Pseudo-Anosov Maps and Eigenfunctions
Abstract: In joint work with Nicolas Bedaride, Pat Hooper, and Pascal Hubert, we use ideas of Furstenberg and Veech to connect results about weak mixing properties of flows on translation surfaces to ergodic properties of linear flows on translation prisms, and use this to obtain several results about unique ergodicity of these prism flows and related billiard flows. We construct explicit eigenfunctions for translation flows in pseudo-Anosov directions with Pisot expansion factors, and use this construction to build explicit examples of non-ergodic prism flows, and non-ergodic billiard flows in right prisms over regular n-gons for n=7,9,14,16,18,20,24,30. In this talk, we will focus on the explicit construction, and speculate on potential connections to Cannon-Thurston maps, inspired by some similarities in pictures.
Note: Jayadev is also giving a colloquium talk on this day at 4:15 in the math commons room.
September 29: No seminar
October 6: No seminar
October 13: No seminar (CUNY on Monday schedule)
October 20: Casey Whitney (CUNY, Graduate Center)
Title: Shear coordinates for infinite type surfaces with a bounded ideal triangulation
Abstract: Given an ideal triangulation of a Riemann surface, there is a relationship between adjacent triangles called the shear. Two adjacent triangles on the surface lift to adjacent ideal triangles in H. The shear between two ideal triangles in H is the signed displacement between the orthogonal projections of the non-shared vertices onto the shared edge. Quasiconformal maps change the shears between adjacent triangles, up to homotopy. We wish to identify what types of surfaces admit such a triangulation and which changes of shear coordinates correspond to quasiconformal deformations. We show that for infinite type punctured surfaces that admit a bounded ideal triangulation, there exists a real analytic parametrization of Teichmuller space by these shear coordinates. This is joint work with Dragomir Saric.
October 27: Dongryul Kim (IAS)
Title: Invariant measures for subgroups of mapping class groups
Abstract: The Thurston measure on the space of measured laminations on a hyperbolic surface is a Radon measure invariant and ergodic under the action of the mapping class group. Lindenstrauss--Mirzakhani and Hamenst\"adt independently classified all invariant Radon measures for the full mapping class group, proving that the Thurston measure is the unique ergodic measure on the subspace of recurrent measured laminations. In this talk, we extend this unique ergodicity on the recurrent measured laminations to "subgroups" of the mapping class group. As a special case, we deduce that for a convex cocompact subgroup, any ergodic invariant Radon measure is either the unique one on the recurrent measured laminations or a counting measure on the orbit of a non-recurrent measured lamination. This is joint work with Inhyeok Choi.
Note: Dongryul will also give a talk in the Geometry and Topology seminar at 4:15pm. The talks are independent.
November 3: Nicholas Miller (Villanova U.)
Title: Geodesic planes in infinite volume
Abstract: In joint work with Bader, Fisher, and Stover, we showed that a finite-volume hyperbolic manifold is arithmetic if it contains infinitely many maximal properly immersed totally geodesic submanifolds of dimension at least two. I will discuss an infinite-volume analogue which shows that a convex cocompact hyperbolic manifold with Fuchsian ends covers an arithmetic hyperbolic manifold if infinitely many such submanifolds meet its convex core; this answers a question of Oh. The talk will develop the motivation for both results from the ground up before turning to some of the geometric and dynamical ideas behind the proof.
November 10: Chenxi Wu (U. of Wisconsin)
Title: Gaps between homological spectral radius and growth rate for irreducible train tracks
Abstract: In a paper by McMullen in 2013, he proved that a pseudo anosov surface homeomorphism can either be lifted to a finite cover where the stretch factor equals the spectral radius of the induced map on homology, or, there is a non zero positive number epsilon, depending only on the original pseudo-Anosov map, such that any lift of the map to an open cover will have spectral radius no more than the stretch factor minus epsilon. This can be interpreted as a property on outer automorphisms of surface groups, and one can ask analogous questions for other hyperbolic groups, and the case of free groups have been raised as a conjecture previously. We resolved this conjecture in the affirmative, proving an analogous theorem in the case of irreducible train tracks maps on finite graphs.
November 17: Abdul Zalloum (Queens U.)
Title: Recognizing convergence actions
Abstract: An action of a group on a perfect, compact, metrizable topological space M is said to be a convergence action if the induced action on the space of distinct triples of M is properly discontinuous. I will discuss recent joint work with Ding in which, relying merely on B.I. (a.k.a. Bowditch's intelligence), we show that every convergence action of a group G on M is induced by a proper isometric action on a hyperbolic space X whose Gromov boundary is M. This answers a question by Misha Kapovich.
November 24: No seminar (Thanksgiving week)
December 1: Cormac O'Sullivan (CUNY, Graduate Center and Bronx Community College)
Title: Patterns in the Markov numbers
Abstract: The Markov numbers 1, 2, 5, 13, 29, 34, 89, 169, ... were originally found in connection with Diophantine approximation in number theory, and later seen in geometry in the lengths of closed geodesics on a particular punctured torus. Bowditch exploited this connection to give a simple proof of McShane's identity for any once-punctured torus. On one side is the multiset of lengths of closed geodesics and on the other is a combinatorial tree structure that displays the Vieta involution linked solutions of x^2+y^2+z^2-xyz=D for D=0. McShane, Rivin and Fock found a remarkable way to encode all of these lengths/solutions with a continuous convex function. In this talk we see how this fits into a bigger picture when D is any real number, and use these ideas to explain basic patterns in these numbers. We are working on the easier combinatorial tree side and hope this can be of use to geometers. See my recent arXiv preprint.
December 8: Ming Hong Tee (Boston College)
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
Matthew Stover (CUNY, Graduate Center) Email: mstover@gc.cuny.edu
Nick Vlamis (CUNY, Graduate Center and Queens College) Email: nvlamis@gc.cuny.edu
HISTORY OF THE HYPERBOLIC GEOMETRY SEMINAR : The hyperbolic geometry seminar has been running continuously since Fall 2008. In the early years it ran mostly as a student seminar with some research talks interspersed. The seminar topics span a wide range of areas including hyperbolic and conformal geometry, geometric structures on manifolds, low dimensional topology, geometric group theory, and many others.
Past organizers have been Youngju Kim (2008-2010), Viveka Erlandsson (2010-2013), Chris Arettines (2013-2015), Blanca Marmolejo (2015-2017), and Daniel White (2017-2019).