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.
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
September 16: Tarik Aougab (Haverford College)
Title: Sharp bounds for 1-systems video
Abstract: On a closed orientable surface of genus g, a 1-system is a collection of pairwise non-homotopic simple closed curves that pairwise intersect at most once. Obtaining bounds on the maximum size of a 1-system has proved to be a surprisingly hard problem. Constructions with roughly g^2 curves have been known for the last few decades, but upper bounds are trickier: in 2012, Malestein-Rivin-Theran produced an upper bound that is exponential in g. Przytycki in 2014 improved this to a bound that is O(g^3), and in 2018 Josh Greene improved this to a bound that behaves like g^2*log(g).
Our main result is a quadratic-in-g upper bound, resolving the problem up to explicit multiplicative constants. We achieve this by choosing an appropriate hyperbolic metric and paying careful attention to how certain polygons formed by curves in the 1-system distribute their area over the surface. This represents joint work with Jonah Gaster.
September 23: No Seminar (CUNY Closed)
Title:
Abstract:
September 30: Matthew Durham (Hunter College)
Title: Asymptotically CAT(0) metrics, Z-structures, and the Farrell-Jones Conjecture video
Abstract: will discuss recent work with Minsky and Sisto, in which we prove that mapping class groups of finite-type surfaces---and more generally, colorable hierarchically hyperbolic groups (HHGs)---are asymptotically CAT(0). This is a simple but powerful non-positive curvature property introduced by Kar, roughly requiring that the CAT(0) inequality holds up to sublinear error in the size of the triangle.
We use the asymptotically CAT(0) property to construct visual compactifications for colorable HHGs that provide Z-structures in the sense of Bestvina and Dranishnikov. It was previously unknown that mapping class groups are asymptotically CAT(0) and admit Z-structures. As an application, we prove that many HHGs satisfy the Farrell-Jones Conjecture, providing a new proof for mapping class groups (Bartels-Bestvina) and establishing the conjecture for extra-large type Artin groups.
To construct asymptotically CAT(0) metrics, we show that every colorable HHG admits a manifold-like family of local approximations by CAT(0) cube complexes, where transition maps are cubical almost-isomorphisms.
October 7: Samuel Taylor (Temple University)
Title: Transverse surfaces and pseudo-Anosov flows video
Abstract: I’ll discuss joint work with Michael Landry and Yair Minsky that characterizes the surfaces in a compact 3-manifold M that are (almost) transverse to a fixed transitive pseudo-Anosov flow. Our main tool is a general correspondence between surfaces that are almost transverse to the flow and those that are “relatively carried” by any associated veering triangulation. The correspondence also allows us to investigate the uniqueness of almost transverse position, to extend Mosher's Transverse Surface Theorem to the case with boundary, and more generally to characterize when relative homology classes represent Birkhoff surfaces of the flow.
October 14: No Seminar (CUNY on Monday Schedule)
October 15: (Special Lecture) Wednesday, 4:00pm. Sara Maloni Colloquium (math commons room)
Title: Geometric Structures in Higher Teichmüller Theory
Abstract: The Teichmüller space of a surface S is the deformation space of marked hyperbolic structures. This can be viewed as a component of representations of the fundamental group π1(S) into the isometry group of hyperbolic space. Higher Teichmüller Theory generalizes this idea by studying representations of surface groups into Lie groups of higher rank.
In this talk, we will introduce key ideas in the theory of deformations of geometric structures, (higher) Teichmüller Theory and the concept of Anosov representations. We'll also discuss recent work with Alessandrini, Tholozan, and Wienhard and work of Mason Hart, showing how these representations also relate to deformations of geometric structures.
October 21: Lorenzo Ruffoni (Binghamton University)
Title: Negatively curved 3-pseudomanifold groups that cannot be cubulated video
Abstract: The fundamental group of a closed hyperbolic 3-manifold is known to act geometrically on a CAT(0) cube complex. We ask whether the same is true for the fundamental group of negatively curved 3-pseudomanifolds, i.e., 3-manifolds with isolated singularities. While many 3-pseudomanifolds are cubulated, such as those arising from RACGs and strict hyperbolization, in this talk we give the first examples of closed 3-pseudomanifolds that are locally CAT(-1) but whose fundamental group cannot be cubulated. These examples are obtained from certain compact hyperbolic 3-manifolds with totally geodesic boundary by coning off the boundary components. This is joint work with J. Manning.
October 28: (No seminar)
November 4: (No seminar)
November 11: Richard Canary (U. of Michigan)
Title: Bending, Entropy and proper affine actions of surface groups video
Abstract: The entropy of a quasifuchsian group agrees with the Hausdorff dimension of its limit and the entropy gives rise to an analytic function on the space of marked quasifuchsian groups. We find an unbounded open neighborhood of the Fuchsian locus in quasifuchsian so that the only critical points of the entropy function lie on the Fuchsian locus.
We also find an open neighborhood of the Fuchsian locus so that (the adjoint of) any quasifuchsian group in the neighborhood arises as the linear part of a proper affine action of the surface group on the Lie algebra of SL(2,C).
Both of these results are obtained by studying the infinitesmal behavior of bending deformations of quasifuchsian groups. This is joint work with Martin Brtidgeman and Andres Sambarino.
November 18: Zhenghao Rao (Rutgers)
Title: Incompressible surfaces in closed hyperbolic manifolds
Abstract: Zhenghao will be giving two talks on incompressible surfaces in closed hyperbolic manifolds, one in the hyperbolic geometry seminar and the other in the CUNY geometry and topology seminar - the second talk will be self-contained and not depend on the earlier talk.
November 25: No Seminar (Thanksgiving week)
Title:
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December 2: Charles Bordenave (IAS)
Title: Cutoff for geodesic paths on hyperbolic manifolds
Abstract: For an ergodic dynamical system, the cutoff describes an abrupt transition to equilibrium. Historically, it was introduced in seminal works of Diaconis, Shahshahani and Aldous for card shuffling and other random walks on finite groups. There are however very examples in non-discrete settings. We establish new instances of the cutoff phenomenon for geodesic paths and for the Brownian motion on compact hyperbolic manifolds. We prove that for any fixed compact hyperbolic manifold, the geodesic path started on a spatially localized initial condition exhibits cutoff. Our work also extends results obtained by Golubev and Kamber on hyperbolic surfaces of large volume to any dimension. More generally, we will discuss ongoing works on the cutoff phenomenon in mixing dynamical systems. This is a joint work with Joffrey Mathien available here https://arxiv.org/abs/2502.06325.
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 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).