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
Title: Domains of discontinuity for Anosov representations
Abstract: The parameter spaces of Anosov representations can usually be seen as deformation spaces of parabolic geometric structures on some closed manifolds, this is due to Guichard-Wienhard and Kapovich-Leeb-Porti. A particularly interesting case is given by the Higher Teichm\"uller Spaces, spaces that generalize the classical Teichm\"uller spaces for higher rank Lie groups.
In this talk we will present a general structure theorem about the topology of such closed manifolds, and we will describe this topology explicitly in some interesting cases.
Title: $C^2$-Morse functions on $\overline{\mathcal M}_{g,n}$ and a distribution theorem on low index critical points
Abstract: In 2003, Akrout proved that the systole function is topologically Morse on the moduli space $\mathcal M_{g,n}$ of Riemann surfaces. However, classical Morse theory cannot be applied. In this talk, we construct a series of $C^2$-Morse functions on the Deligne-Mumford compactification of $\mathcal M_{g,n}$ that converges to systole and compare critical points of these functions to those for systole. We show that all low index critical points for these functions exist in the Deligne-Mumford boundary $\partial\mathcal M_{g,n}$ for large g or n, which implies all low degree handles in the Morse handle decomposition appear in the boundary. We also give a classification of all index 0, 1 and 2 critical points.
Title: Big mapping class groups that are small
Abstract: The mapping class group of a 2-manifold is big if the 2-manifold is of infinite type (i.e. its fundamental group is not finitely generated). Big mapping class groups are big in the sense that they are uncountable groups and, as topological groups, are not locally compact. Despite this, they can be geometrically small. We will discuss various ways that a group may be geometrically small and the various classes of big mapping class groups that satisfy these smallness conditions. Some of the work discussed is joint with Justin Lanier.
Title: Horizontal foliation of integrable holomorphic quadratic differentials on infinite Reimann surfaces
Abstract: Let $X$ be an infinite Riemann surface with the covering group of the first kind. We prove that the Brownian motion on $X$ is recurrent iff a.e. horizontal leaf of every integrable holomorphic quadratic differential on $X$ is recurrent. When $X$ is such, we prove that the holomorphic quadratic differentials with single horizontal cylinders (the Jenkins-Strebel differentials) are dense among all integrable holomorphic quadratic differentials in the L^1-norm. We also extend Kerckhoff’s formula for the Teichmuller distance in terms of the extremal lengths of simple curves for such Riemann surfaces.
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Title: Complex hyperbolic and projective deformations of Kleinian groups
Abstract: We consider deformations of discrete subgroups (in particular lattices) of SO(3,1) into the larger Lie groups SU(3,1) and SL(4,R). In particular, when such deformations exist we would like to know whether or not they remain discrete and faithful in some neighborhood of the inclusion. We will review results of Cooper-Long-Thistlethwaite andBallas-Danciger-Lee in the manifold case, then discuss recent joint work with Morwen Thistlethwaite for certain Bianchi groups.
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Title: Distinguishing filling curve types via special metrics
Abstract: In this talk, we look at filling curves on hyperbolic surfaces and consider its length infima in the moduli space of the surface as a type invariant. In particular explore the relations between the length infimum of curves and their self-intersection number. For any given surface, we will construct infinite families of filling curves that cannot be distinguished by self-intersection number but via length infimum. I might also discuss some coarse bounds on the metrics associated to these minimum lengths.
Title: Volume and homology of hyperbolic 3-manifolds
Abstract: If M is a closed, orientable hyperbolic 3-manifold, the Mostow rigidity theorem implies that any geometric invariant of M, such as its volume, is a topological invariant. Much of my research over the last 30 years has focused on establishing explicit connections between geometrically defined invariants and more classical topological invariants. I will illustrate the very rich and varied methods involved in this project by discussing the proof of a recent result, proved jointly with Jason DeBlois, which asserts that if the volume of M is at most V_oct / 2, where V_oct = 3.66... is the volume of a regular ideal hyperbolic octahedron, then dim H_1(M;F_2) is at most 4. This result is probably not sharp, because the largest value of dim H_1(M;F_2) that is known to occur when the volume in this range is 2; but the result is in the right ballpark. The proof involves a combination of geometric analysis, 3-manifold topology, algebraic topology and classical geometry.
Title: A Bers type classification of big mapping classes
Abstract: We show that the space of marked hyperbolic structures with the same ending geometry on an infinite type surface $\Sigma$ is locally path connected, connected with respect to the Fenchel-Nielsen topology. In particular, the space of marked complete structures is locally path connected, connected. Since having the same ending geometry is a quasiconformal (qc) invariant, the space of such structures decomposes into the disjoint union of Teichmuller subspaces.
We study the action of the mapping class group on these Teichmuller subspaces and, in particular, show that there are classes that are never qc.
Title: Growth of Conjugacy Classes of Reciprocal Words in Triangle Groups
Abstract: A (p,q,r) - triangle group is a group containing an order p, order q, and order r element where 1/p +1/q + 1/r <1. A fundamental domain for a triangle group is a quadrilateral formed from two hyperbolic triangles both of which have internal angles of $\pi/p, \pi/q,$ and $\pi/r$. We focus here on Hecke groups G, i.e. the (2,q,\infty) - triangle groups. On the orbifold which arises from the quotient by the action of G on the hyperbolic plane, we study closed geodesics called reciprocal geodesics, which pass through the order two cone point. To understand these special geodesics, we consider the corresponding reciprocal words in the group. In particular, we obtain the growth rates for conjugacy classes of primitive reciprocal words in a class of Hecke groups. We use combinatorial methods and recurrence relations to count these geometric objects and show the asymptotic behavior of the conjugacy classes with respect to word length.
History of the hyperbolic geometry seminar