This is the main page for the Joint Seminar on Teichmüller theory and Related topics (JSTeichR), organized by:

- Yi HUANG 黄意 (Yau Mathematical Sciences Center, Tsinghua University)
- Qiongling LI 李琼玲 (Chern Institute of Mathematics, Nankai University)
- Yi LIU 刘毅 (Beijing International Center of Mathematical Research, Peking University)
- Yunhui WU 吴云辉 (Yau Mathematical Sciences Center, Tsinghua University)

Here's the attendance information for the seminar series:

- Time: Tuesday, 10:00 AM (Beijing time).
- Place: online.
- Zoom: 405 416 0815, pw: 111111

Brooks and Makover developed a combinatorial model of random hyperbolic surfaces by gluing certain hyperbolic ideal triangles. In this talk we show that for any \(\epsilon>0\), as the number of ideal triangles goes to infinity, a generic hyperbolic surface in Brooks-Makover's model has Cheeger constant less than \(\frac{3}{2\pi}+\epsilon\). This is a joint work with Yunhui Wu.

The moduli space \(M_g\) of Riemann surfaces has many different incarnations, each of which equips it with different geometric structures. Thinking about \(M_g\) as the space of complex structures gives rise to the well-studied horocycle flow and the field of Teichmüller dynamics, while thinking about it as the space of hyperbolic metrics yields the much more mysterious earthquake flow. In this talk, I’ll discuss to what extent these flows are “the same” and how this connection can be used to address a hyperbolic-geometric conjecture of Mirzakhani using flat geometry. This represents joint work (some of which is in progress) with James Farre.

In this talk, we will introduce Riemannian metrics, called pressure metrics, and Finsler metrics, called Thurston's asymmetric metrics in higher Teichmüller spaces. Both are generalization of classical metrics on Teichmüller spaces and are constructed using Thermodynamic formalism. The pressure metrics were first studied in works of Bridgeman, Canary, Labourie and Sambarino. The Thurston's asymmetric metrics were recently introduced in joint work with Leon Carvajales, Beatrice Pozzetti and Anna Wienhard.

- Seminar notes.
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Kahn and Markovic proved the Surface Subgroup conjecture for compact hyperbolic 3-manifolds more than ten years ago. And the surface subgroup they constructed can be as close as possible to Fuchsian. But a compact hyperbolic 3-manifold can also have surface subgroups far away from being Fuchsian. Actually, provided any genus-2 quasi-Fuchsian group \(\Gamma\) and cocompact Kleinian group \(G\), we can show that for any \(K>1\), one can find a surface subgroup \(H\) of \(G\) that is \(K\)-quasiconformally conjugate to a finite index subgroup \(F<\Gamma\). As an application of this result, we can prove that the set of Hausdorff dimensions of limit sets of surface subgroups of \(G\) is dense in \([1,2]\).

Gromov boundary provides a useful compactification for all infinite-diameter Gromov hyperbolic spaces. It consists of all geodesic rays starting at a given base-point and it is an essential tool in the study of the coarse geometry of hyperbolic groups. In this study we introduce a generalization of the Gromov boundary for all finitely generated groups called the sublinearly Morse boundaries. We will show that it is a QI-invariant topological space and it is metrizable. It is the first such topology boundary for all proper geodesic metric spaces.

To argue that the sublinearly Morse boundary is natural and large, we show that the sublinearly Morse directions are generic in several senses: they are generic with respect to natural geometric measures on the boundaries of rank-1 CAT(0) spaces and the Teichmüller space; and they are generic in the random walk measure on the Poisson boundary of several classes of groups. If time permits, we will discuss possible compactification of this boundary and other applications. The construction of the boundary is joint with Kasra Rafi and Giulio Tiozzo.

The trace functions of multicurves form a linear basis of \(\mathrm{SL}_2\) character variety of surface group. The \(\mathrm{SL}_3\) analogue of the multicurves are the \(\mathrm{SL}_3\) webs. The Fock-Goncharov-Shen duality in this case says that the tropical integral points of decorated punctured \(\mathrm{SL}_3\) character variety parameterize the canonical linear basis of the regular function ring of the \(\mathrm{SL}_3\) character variety. In this talk, firstly I will explain my joint work with Daniel Douglas where we relate both sides to the \(\mathrm{SL}_3\)-webs. Then I will explain my joint work with Linhui Shen and Daping Weng, where we give a topological intersection number for webs which allows us to prove the mutation equivarience of the bijection between webs and tropical points.

Teichmüller space is a simply connected manifold that admits an action of the mapping class group. It is known that there exist mapping class group-equivariant deformation retractions of Teichmüller space onto cell complexes (spines) of nonzero codimension. The virtual cohomological dimension of the mapping class group is approximately two thirds of the dimension of Teichmüller space and gives a lower bound on the dimension of such a spine. This talk will prove that this lower bound is realised for Teichmüller spaces of closed, orientable, hyperbolic surfaces.

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Labourie proved that every Hitchin representation into \(\mathrm{PSL}(n,\mathbb{R})\) gives rise to an equivariant minimal surface in the corresponding symmetric space. He conjectured that uniqueness holds as well (this was known for \(n=2,3\), and explained that if true, then the Hitchin component admits a mapping class group equivariant parametrization as a holomorphic vector bundle over Teichmüller space. After giving the relevant background, we will explain that Labourie’s conjecture fails for n at least 4, and point to some future questions.

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Hitchin's theory of Higgs bundles associated holomorphic differentials on a Riemann surface \((S,J)\) to representations of the fundamental group of the surface \(S\) into a Lie group. We study, in a pair of very low rank settings, the geometry common to representations whose associated holomorphic differentials are on a ray of differentials. In the setting of \(\mathrm{SL}(3,\mathbb{R})\), we provide a formula for the asymptotic holonomy of the representations in terms of the local geometry of the differential; the formula has a tropical flavor which we do not develop. Alternatively, we show how the associated equivariant harmonic maps to a symmetric space converge to a harmonic map to a building, with geometry determined by the differential. All of this is joint work with John Loftin and Andrea Tamburelli.

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In 1976, S. Patterson introduced a class of conformal measures on the limit set of Fuchsian groups, and further developed by D. Sullivan in Kleinian groups with various applications in spectral geometry, Mostow rigidity, geodesic flows and complex dynamics etc. These now called Patterson-Sullivan measures are important instances of conformal density. In this talk, we will explain how to establish a theory of Patterson-Sullivan measures on the Thurston and Gardiner-Masur boundaries for mapping class groups. In particular, the Shadow Lemma and Hopf-Tsuji-Sullivan theorem are obtained for any non-elementary subgroups. Our approach is general and works for any discrete group action on metric spaces with contracting elements.

A famous theorem of Fermat says that a prime number \(p\) can be written as a sum of squares \(a^2 + b^2\) if and only if \(p\) is 2 or \(p-1\) is divisible by \(4\).

In the 80s R. Heath-Brown gave a proof of this result by studying the action of a group of order 4 on a finite set.

We'll explain another proof of this result based on the automorphisms of the 3 punctured sphere. An important ingredient in our approach is the use of Penner's lambda lengths.

We'll also show how this is related to Button's partial result on the Markoff unicity conjecture: a Markoff number is unique if it is prime.

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Cutting a hyperbolic surface along a simple closed geodesic yields a hyperbolic structure on the complementary subsurface. We study the distribution of the shapes of these subsurfaces in moduli space as boundary lengths go to infinity, showing that they equidistribute to the Kontsevich measure on a corresponding moduli space of metric ribbon graphs. In particular, random subsurfaces look like random ribbon graphs, a law which does not depend on the initial choice of hyperbolic surface. This result strengthens Mirzakhani’s famous simple close multi-geodesic counting theorems for hyperbolic surfaces. This is joint work with Aaron Calderon.

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An abelian differential induces a flat metric with conical singularities such that the underlying surface can be realized as a polygon whose edges are pairwise identified via translations. Period coordinates arising from edges of the polygons induce a volume form on moduli spaces of abelian differentials with prescribed cone angles, called the Masur—Veech volume form. This talk will introduce various approaches to compute Masur—Veech volumes of moduli spaces of abelian differentials, with a focus on the interplay between algebraic geometry, analytic geometry, combinatorics, and Teichmüller dynamics.

In this talk, I am going to talk about the sparseness of Thurston's subset. Sparseness is a geometric concept on Thurston's subset firstly raised by Anderson-Parlier-Pettet in 2016. We have proved the sparseness of Thurston's subset in the sense of Teichm\"uller distance and Weil-Petersson distance. More precisely, most surfaces in genus g surface moduli space have Teichmüller distance \(\frac{1}{5}\log\log g\) and Weil-Petersson distance \(0.6521(\sqrt{\log g}-\sqrt{7\log\log g})\) to the Thurston's subset. Some recent progresses on random surface (Mirzakhani-Petri, Nie-Wu-Xue) and estimate of the Weil-Petersson distance by systole (Wu) are important tools in our proof.

Teichmüller space admits several ray structures. In this talk, we first depict harmonic map ray structures on Teichmüller space as a geometric transition between Teichmüller ray structures and Thurston geodesic ray structures. In particular, by appropriately degenerating the source of a harmonic map between hyperbolic surfaces (along "harmonic map dual rays"), the harmonic map rays through the target converge to a Thurston geodesic; by appropriately degenerating the target of the harmonic map, those harmonic map dual rays through the domain converge to Teichmüller geodesics. While there may be many Thurston metric geodesics between a pair of points in Teichmüller space, we find that by imposing an additional energy minimization constraint on the geodesics, thought of as limits of harmonic map rays, we select a unique Thurston geodesic through those points. Extending the target surface to the Thurston boundary yields, for each point Y in Teichmüller space, we construct an "exponential map" of rays from that point Y onto Teichmüller space with visual boundary the Thurston boundary of Teichmüller space. This is a joint work with Michael Wolf (arXiv:2206.01371).

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In this talk, I will survey what is known or conjectured about maps between configuration spaces and moduli spaces of surfaces. Among them, we consider two categories: continuous maps and also holomorphic maps. We will also talk about some results and conjectures about maps between finite covers of those spaces. Unlike the case of the whole spaces, much less is known about their finite covers.

The universality of Brownian motion underlies the appearance of SLE in conformally invariant 2D systems. The action functional of Brownian motion, namely the Dirichlet energy, corresponds to the Loewner energy for a Jordan curve. This energy, intuitively speaking, measures the roundness of a Jordan curve and the density of SLE around the curve. Furthermore, this energy is shown by Takhtajan and Teo to be a Kahler potential on the space of Weil-Petersson quasicircles, identified with the Weil-Petersson universal Teichmüller space. I will overview different geometric and probabilistic descriptions of the Loewner energy and Weil-Petersson Teichmüller space and discuss further development by exploring those links.

I will talk about mapping class group action on the curve complex with the following conjecture in mind: an element with small stable translation length is a normal generator. This conjecture is motivated by a similar statement in the case of the action on the Teichmüller space proved by Lanier-Margalit. I will discuss various partial results (based on joint works with various subsets of \(\{\)Dongryul Kim, Hyunshik Shin, Philippe Tranchida, Chenxi Wu\(\}\)).

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Spacelike minimal surfaces in the pseudo-hyperbolic (2+q)-spacetime are actually maximal and are related to \(\mathrm{SO}\)(2,q+1)-representations of surface groups through the works of Bonsante-Schlenker and Collier-Tholozan-Toulisse. We investigate a type of minimal surface associated to \(\mathrm{SO}\)(n,n+1)-representations. They occur in pseudo-hyperbolic spacetimes with higher space dimensions, and hence are not maximal; but they are still infinitesimally rigid, hence can be used to prove Labourie's conjecture in rank \(2\).

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Let \(G\) be a Lie group acting transitively on a manifold \(X\). A *compact quotient of \(X\)* (also called *compact Clifford--Klein form*) is simply a quotient of \(X\) by a discrete subgroup of \(G\) acting properly discontinuously and cocompactly. In this talk, we will address the following questions:

- Does \(X\) admit compact quotients?
- If so, what is their topology?
- If so, do these compact quotients have deformations?

These questions are rooted in the mathematics of the late XIX^{th} century: Klein's *Erlangen program*, the uniformization of (closed) Riemann surfaces, Poincaré's *Analysis Situs*...

If there are any issues, please feel free to contact any of the organizers.