February 9th: Priyam Patel (Purdue University)
Title: Lifting curves simply in finite covers
Abstract: It is a well known result of Peter Scott that the fundamental groups of surfaces are subgroup separable. This algebraic property of surface groups also has important topological implications. One such implication is that every immersed (self-intersecting) closed curve on a surface lifts to an embedded (simple) one in a finite cover of the surface. A natural question that arises is: what is the minimal degree of a cover necessary to guarantee that a given closed curve lifts to be embedded? In this talk we will discuss various results answering the above question for hyperbolic surfaces, as well as several related questions regarding the relationship between geodesic length and geometric self-intersection number. Some of the work that will be presented is joint with T. Aougab, J. Gaster, and J. Sapir.
February 16th: Kenneth Bromberg (University of Utah)
Joint meeting with CUNY Geometry and Topology Seminar. Please note this meeting is in room 3212 at 4:15pm.
Title: Stable commutator length in the mapping class group
Abstract: The mapping class group is an example of a perfect group; its abelianization is trivial. In particular, every element can be written as a product of commutators. Endo and Kotschik showed that the mapping class group is not uniformly perfect; there is no bound on the number of commutators required to represent a given element. To prove this they showed that there are elements with positive ``stable commutator length.'' Their proof uses rather sophisticated results on the symplectic geometry of 4-manifolds. In this talk we will use more elementary methods to give a complete characterization of when the stable commutator length is positive in the mapping class group. The is joint work with M. Bestvina and K. Fujiwara.
Link to CUNY Geometry and Topology Seminar: Geometry and Topology Seminar
February 23rd: No meeting (Scheduling conflict)
March 1st: Dragomir Saric (CUNY Graduate Center and Queens College)
Title: Thurston's boundary to infinite Teichmüller spacees
Abstract: We introduce Thurston's boundary to Teichmüller space of an arbitrary hyperbolic Riemann surface using the notion of geodesic currents. In the case of a Riemann surface with an infinitely generated fundamental group, we give an alternative definition using the length spectrum. Finally, we compare the two boundaries.
March 8th: Daniel Gallo (St. John's University)
Title: Characterizing hyperelliptic surfaces
Abstract: A closed Riemann surface S of genus g ≥ 2 is hyperelliptic if it has a conformal involution J: S -> S with 2g + 2 fixed points. Following results of Schmutz-Schaller and, separately, Maskit, such a surface can be characterized by the existence of certain sets of simple, closed geodesics on the surface. In this talk we give a generalized characterization in terms of simple, closed geodesics and graphs associated to these geodesics.
March 15th: Carolyn Abbott (University of Wisconsin- Madison)
Title: An introduction to acylindrically hyperbolic groups
Abstract: Acylindrical hyperbolic groups are a recent generalization of hyperbolic groups, defined by Denis Osin in 2013. They are a broad class of groups, including many groups that were of independent interest, such as mapping class groups, Out(Fn), and many fundamental groups of 3-manifolds. Despite including such a wide variety of groups, the class is narrow enough to have a non-trivial theory. In this talk, we will give multiple definitions of acylindrically hyperbolic groups and discuss some basic properties.
March 22nd: No meeting (Scheduling conflict)
March 29th: Dragomir Saric (CUNY Graduate Center and Queens College)
Please note this meeting begins at 2:45pm.
Title: The length spectrum metric on infinite-dimensional Teichmüller spaces
Abstract: We discuss certain aspects of the length spectrum distance on infinite-dimensional Teichmüller spaces. The topics will include the Fenchel-Nielsen coordinates, the Thurston-type boundary, and type problems.
April 5th: Yan Mary He (University of Chicago)
Title: Basmajian-type identities and Hausdorff dimension of limit sets
Abstract: In this talk, I will discuss Basmajian-type series identities on holomorphic families of Cantor sets associated to familiar one-dimensional complex dynamical systems. In particular, I will show how to extend Basmajian's identity to certain Schottky groups via analytic continuation and exhibit examples of nontrivial monodromy. Moreover I will introduce Basmajian-type identities for complex quadratic polynomials.
April 12th: Ser-Wei Fu (Temple University)
Title: Cusp excursions of earthquake flows
Abstract: The earthquake map introduced by Thurston describes 1-parameter deformation families in the space of hyperbolic surfaces. The earthquake flow on the bundle over moduli space is shown to be ergodic by Mirzakhani. This motivates further research on the ``generic'' behavior of the flow-lines. In this talk, I will briefly overview the properties of the earthquake flow and describe the cusp excursion problem. In particular, I will prove an almost everywhere statement regarding to the divergence speed, given in the sense of the systole length, of flow lines of the earthquake flow over the once-punctured torus. The result will be compared to similar statements for Teichmüller geodesic flows and horocycle flows.
April 19th: Meeting cancelled
April 26th: No meeting (Spring recess)
May 3rd: Nicholas Vlamis (University of Michigan)
Title: Basmajian's identity in higher Teichmüller theory
Abstract: Basmajian's original identity gives the area of the boundary of a compact hyperbolic manifold as a summation over the orthospectrum. We will demonstrate an extension of this identity to the setting of Hitchin representations of surface groups. We will see that for 3-Hitchin representations, the identity has a natural geometric interpretation analogous to the hyperbolic setting. A Hitchin representation of a closed surface gives rise to a Frenet curve in projective space, which supports a Lebesgue measure. As part of our proof, we will show that the limit set of an incompressible subsurface of a closed surface has null measure with respect to this Lebesgue measure. This generalizes a classical result in hyperbolic geometry. Time permitting, we will introduce a relationship between Basmajian's identity and the McShane-Mirzakhani identity in both the hyperbolic and Hitchin settings. This is joint work with Andrew Yarmola.
Diversity and Inclusion at the GC