Hyperbolic geometry and manifolds: Curves and surfaces

Counting simple closed multi-geodesics on hyperbolic surfaces with respect to the lengths of individual components- Francisco Arana-Herrera (Stanford)

Abstract: In her thesis, Mirzakhani showed that on any closed hyperbolic surface of genus g, the number of simple closed geodesics of length at most L is asymptotic to a polynomial in L of degree 6g-6. Wolpert conjectured that analogous results should hold for more general countings of multi-geodesics that keep track of the lengths of individual components. In this talk we will present a proof of this conjecture which combines techniques and results of Mirzakhani as well as ideas introduced by Margulis in his thesis.

Cutting and pasting along measured laminations- Aaron Calderon (Yale)

Abstract: One of the fundamental techniques of low-dimensional topology is cutting and pasting along embedded codimension 1 submanifolds. In surface theory these submanifolds are just simple closed curves, and cutting and pasting gives rise to Fenchel-Nielsen coordinates for Teichmüller space, normal forms for simple closed curves, and many other foundational results. The set of simple closed curves completes to the space of “measured laminations,” and in my first talk, I will summarize how to build coordinates for Teichmüller space by cutting and pasting along any lamination, generalizing the shear coordinates of Bonahon and Thurston. In my second talk, I will explain how these coordinates lead to an extension of Mirzakhani’s conjugacy between the earthquake and horocycle flows, two notions of unipotent flow coming from hyperbolic, respectively flat, geometry. This represents joint work with James Farre.

Volumes and filling collections of multicurves- Andrew Yarmola (Princeton)

Abstract: Consider a link L in a Seifert-fibered space N over a surface S of negative Euler characteristic. If the fiber-wise projection of L to S is a collection C of closed curves in minimal position, then N \ L is hyperbolic if and only if C is filling and N \ L is acylindrical. The behavior of vol(N \ L) in terms of the topology and geometry of C have been studied in recent years, but effective lower bounds have been elusive. In this talk we will focus on the case where C is a collection of simple closed curves. In the special case where N = PT(S) is the projectivized tangent bundle and L is the canonical lift of a pair of filling multicurves, we show that vol(N \ L) is quasi-isometric to the Weil-Petersson distance between the corresponding strata in Teichmuller space. In the more general setting, we show that vol(N \ L) is quasi-isometric to expressions involving distances in the pants graph whenever L is a stratified link. This is joint work with T. Cremaschi and J. A. Rodriguez-Migueles.

4 replies on “Hyperbolic geometry and manifolds: Curves and surfaces”

Hi Francisco,

Thanks for the very clear and intuitive explanation of some of your work. I have a question about the importance of passing to the intermediate cover to get rid of the redundancy in defining the relevant measures in the final quotient:

– Instead of thinking about putting a measure supported on a horoball in (a bundle over) Teichmuller space, and pushing that forward to the moduli space, can we define a measure as follows?

Choose an arbitrary basepoint in that horoball upstairs and start expanding via symmetric Thurston geodesic flow (or something), always intersecting with the horoball (maybe also do this in ML x T_g to get the correct density function). Then push this Lebesgue measure on this all the way down to and divide by the multiplicity. Take a limit.

This should give the same measure that you consider, right? In particular, it’s locally finite, which seems to me to be the most important feature.

What are the technical difficulties that this introduces? Maybe in computing certain asymptotics (the W-polynomial?)?


Hi James,

Thanks for your question!

I am not sure exactly what you mean by multiplicity, but I believe the measure you describe is the same measure I consider.

As you mention, a convenient thing about the definition I consider is that one can use Mirzakhani’s integration formulas to compute the total mass these measure assign to moduli space (they are actually finite measures) and their asymptotics as the parameter L goes to infinity. This might be harder to do using your definition.

Hi all, just wanted to post here and say that I’m looking forward to answering any questions, or just chatting more, about my talk! Also happy to talk about whatever else (though that should probably be done on a different board).

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