For the time being, the seminar this semester will continue 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.
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
Our first meeting will be on February 22.
We are leaving the possibility of having the seminar in person in April and May. We will revisit this possibility later in the semester.
Note the special time for this week only. Doors open at 4:10.
Title: A case of pentagram rigidity
Abstract: I will explain a rigidity result that relates to Poncelet polygons and to the pentagram map, a discrete integrable system that I've spent half my life thinking about, off and on. I view the rigidity result as sort of a projective geometry version of circle packing rigidity, though my result is much more specific. The proof involves studying the geometry of a singular Lagrangian foliation in R^4. I'll explain everything from scratch and illustrate the talk with computer demos.
Title: Detecting covers, simple closed curves, and Sunada's construction.
Abstract: Given a pair of finite degree (not necessarily regular) covers (p,X),(q,Y) of a finite type surface S, we show that the covers are equivalent if and only if the following holds: for any closed curve gamma on S, some power of gamma admits an embedded lift to X if and only if some power of gamma admits an embedded lift to Y. We apply this to study the well-known construction of Sunada which yields pairs of hyperbolic surfaces (X,Y) that are not isometric but that have the same unmarked length spectrum. In particular we show that the length-preserving bijection from closed geodesics on X to those on Y arising from the Sunada construction never sends simple closed geodesics to simple closed geodesics. We also show that length-isospectral surfaces arising from several of the most well-known manifestations of the construction are not simple length isospectral. Even more, we construct length-isospectral hyperbolic surfaces so that for each finite n, the set of lengths corresponding to closed geodesics with at most n self intersections disagree. This represents joint work with Maxie Lahn, Marissa Loving, and Nicholas Miller.
Title: Small covers of big surfaces
Abstract: I will talk about work in progress with Alan McLeay investigating the following apparently innocent question: Given two surfaces, when does there admit a finite-sheeted cover of one over the other? A complete answer is available if the two surfaces are of finite type. In the infinite-type world, the question appears to be less innocent than one might expect.
Title: Subsurface projections in fibered 3-manifolds
Abstract: To a fibered hyperbolic 3-manifold one can associate a function on the set of subsurfaces of the fiber, giving the curve-complex projection distances of the stable/unstable foliations in each subsurface. The geometry of the manifold is coarsely related to this function, although the quality of the approximation degrades as the fiber genus grows. Manifolds with (infinitely) many fibrations provide a way to study dependence of this picture on the fiber, which leads one to consider a "fibered face" of Thurston's norm, as studied by Thurston, Fried and many others. We obtain uniformity properties for the "large projections" over all fibrations in a given fibered face. This is joint work with Sam Taylor.
Title: Combinatorial growth in the modular group
Abstract: Consider the modular surface, that is, the (2,3, \infty) triangle orbifold. A reciprocal geodesic on the modular surface is a closed geodesic that begins and ends at the order two cone point, traversing its image twice. In this talk we consider an exhaustion of the modular surface by compact subsurfaces and show that the growth rate, in terms of word length, of the reciprocal geodesics on such subsurfaces (so called low lying reciprocal geodesics) converges to the growth rate of the full set of reciprocal geodesics on the modular surface. We derive a similar result for the low lying geodesics and their growth rate convergence to the growth rate of the full set of closed geodesics. This is joint work with Ara Basmajian.
Title: Asymptotic mapping class groups of Cantor manifolds and their finiteness properties
Abstract: (Joint work with K.-U. Bux, J. Flechsig, N. Petrosyan, X. Wu) A Cantor manifold C is a non-compact manifold obtained by gluing (holed) copies of a fixed compact manifold Y in a tree-like manner. Generalizing braided Thompson groups, we introduce the asymptotic mapping class group of C, whose elements are proper isotopy classes of self-diffeomorphisms of C that are ”eventually trivial.” This group B happens to be an extension of a Higman-Thompson group by a direct limit of mapping class groups of compact submanifolds of C.
B acts on a contractible cube complex X of infinite dimension, and we may try to use the action to determine finiteness properties of B. In well-behaved cases, X is CAT(0) and B is of type F∞; more concretely, the methods apply when Y is a 2-dimensional torus, or S2 × S1, or Sn × Sn for n at least 3. In these cases, the group B contains, respectively, the mapping class groups of every compact surface with boundary, the automorphism groups of finitely generated free groups, or an infinite familiy of arithmetic symplectic or orthogonal groups.
In particular, the cases where Y is a sphere or a torus in dimension 2 yields a positive answer to a question of Funar-Kapoudjian-Sergiescu. In addition, we find cases where the homology of B coincides with the stable homology of the relevant mapping class groups.