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Current contacts: Dave Futer, Matthew Stover, or Sam Taylor.
The Seminar usually takes place on Fridays at 10:40 AM via Zoom (please contact the seminar organizers for the Zoom link), not in Room 617 on the sixth floor of Wachman Hall.
Federica Fanoni, University of Paris
Abstract: Two hyperbolic surfaces are said to be (length) isospectral if they have the same collection of lengths of primitive closed geodesics, counted with multiplicity (i.e. if they have the same length spectrum). For closed surfaces, there is an upper bound on the size of isospectral hyperbolic structures depending only on the topology. We will show that the situation is different for infinite-type surfaces, by constructing large families of isospectral hyperbolic structures on surfaces of infinite genus.
Carmen Galaz-Garcia
University of California, Santa Barbara
Consider a discrete subgroup H of PSL(2,R) and its action on IH^2 the upper half-plane model of the hyperbolic plane. The cusp set of H is the set of points in the boundary at infinity of IH^2 fixed by its parabolic elements. For example, the cusp set of PSL(2,Z) is QU{oo}. A natural question is: how strong is the cusp set as an invariant? More precisely, if H has cusp set QU{oo} , is it commensurable with PSL(2,Z)? A negative answer was provided by Long and Reid in 2001 by constructing finitely many examples of pseudomodular groups. In 2016 Lou, Tan and Vo produced two infinite families of pseudomodular groups via the jigsaw construction. In this talk we will construct a third family of pseudomodular groups obtained with the jigsaw construction and also show that "many" of the simplest jigsaw groups are not pseudomodular.
Sam Taylor
Temple University
Veering triangulations form a rich class of ideal triangulations of cusped hyperbolic manifolds that were introduced by Agol and have connections to hyperbolic geometry, Teichmuller theory, and the curve complex. In this talk, we introduce a polynomial invariant of a veering triangulation that, when the triangulation comes from a fibration, recovers the Teichmuller polynomial introduced by McMullen. We show that in general, the polynomial determines a (typically non-fibered) face of the Thurston norm ball and that the classes contained in the cone over this face have representatives that are carried by the veering triangulation itself.
This is joint work with Michael Landry and Yair Minsky.
Radhika Gupta, Temple University
Didac Martinez-Granado, UC Davis
Abstract: Geodesic currents are measures that realize a closure of the space of curves on a closed surface. Bonahon introduced geodesic currents in 1986, showed that geometric intersection number extends to geodesic currents and realized hyperbolic length of a curve as intersection number with a geodesic current associated to the hyperbolic structure. Since then, other functions on curves have been shown to extend to geodesic currents. Some of them extend as intersection numbers, such as negatively curved Riemannian lengths (Otal, 1990) or word length w.r.t. simple generating sets of a surface group (Erlandsson, 2016). Some other functions aren't intersection numbers but extend continuously (Erlandsson-Parlier-Souto, 2016), such as word length w.r.t. non-simple generating sets or extremal length of curves. In this talk we present a criterion for a function on curves to extend continuously to geodesic currents. This is joint work with Dylan Thurston.
Ian Runnels, University of Virginia
Abstract: Inspired by Ivanov's proof of the Tits alternative for mapping class groups via ping-pong on the space of projective measured laminations, Koberda showed that right-angled Artin subgroups of mapping class groups abound. We will outline an alternate proof of this fact using the hierarchy of curve graphs, which lends itself to effective computations and stronger geometric conclusions. Time permitting, we will also discuss some applications to the study of convex cocompact subgroups of mapping class groups.
Anschel Schaffer-Cohen, University of Pennsylvania
Abstract: Mapping class groups of infinite-type surfaces, also known as big mapping class groups, can be studied geometrically from the perspective of coarsely bounded generating sets. Within this framework, we describe a large family of surfaces--the avenue surfaces without significant genus--and show that the mapping class group of any such surface is quasi-isometric to an infinite-dimensional cube graph. As a consequence, we see that these mapping class groups are all quasi-isometric to each other, and that they are all a-T-menable. Both of these properties are notable in that they are known to fail for mapping class groups of finite-type surfaces.Marissa Loving, Georgia Tech
Abstract: In this talk, I will share some of my ongoing work with Tarik Aougab, Max Lahn, and Nick Miller in which we explore the simple length spectrum rigidity of hyperbolic metrics arising from Sunada’s construction. Along the way we give a characterization of equivalent covers (not necessarily regular) in terms of simple elevations of curves, generalizing previous work with Aougab, Lahn, and Xiao.
Aaron Abrams, Washington & Lee University
Abstract: A celebrated theorem of Monsky from 1970 implies that it is impossible to dissect a square into an odd number of triangles of equal area.2008 | 2009 | 2010 | 2011 | 2012 | 2013 | 2014 | 2015 | 2016 | 2017 | 2018 | 2019 | 2020 | 2021