Talks & Abstracts

Given an essential closed curve on a closed surface of genus at least 2, its inf length is the infimum of its length among all hyperbolic metrics in Moduli space. It is not difficult to see that the inf length remains unchanged in the mapping class group orbit of a curve. By considering the inf length over mapping class group orbits of curves, we are naturally led to a spectrum of non-negative real numbers we call the inf spectrum. The focus of this talk will be on this spectrum.

This is joint work with Sayantika Mondal and Hugo Parlier.

We study the parametrizations and deformations of hyperbolic cone-surfaces of fixed topological type with variable positive cone angles. Inspired by Thurston's horocyclic foliations, we use geodesic triangulations and circular foliations to construct a coordinate atlas for their Teichmuller space. In these coordinates, we introduce peripheral stretch and interior anti-stretch deformations that vary the cone angles in a controlled manner. Their asymptotic limits are, respectively, a cusped hyperbolic metric and a circle-packed hyperbolic cone metric. As an application, we establish sharp upper bounds for admissible cone angles on the universally triangulable locus. This is joint work with Youliang Zhong.

In 1995, McShane and Rivin showed that, on a hyperbolic once-punctured torus, the lengths of simple closed geodesics extend naturally to a norm on homology. Its unit ball gives a convex-geometric encoding of the simple length spectrum. In this talk, I will discuss recent developments arising from this viewpoint, including bounds on simple-length multiplicities, a connection with exponential Diophantine approximation, and new identities for simple closed geodesics.

We introduce flippered hyperbolic surfaces, and study the volumes of their moduli spaces. Particular focus will be given to the case of flippered disks and annuli.

We settle down a Gromov's conjecture that for a non-positively curved 4-manifold with non-zero Euler number, its simplicial volume is positive. We give a few applications of this result. This is a joint work with X. Wan.

In a real hyperbolic 2-manifold, the collar lemma says that a closed geodesic has an embedded tubular neighborhood whose width depends only on the length of the geodesic. In fact, a codimension 1 totally geodesic embedded submanifold in a real hyperbolic $n$-manifold also has such a tubular neighborhood that depends only on its (n-1)-volume. The width of the tubular neighborhood does not depend on the geometry of underlying manifold. On the other hand, a totally geodesic surface with codimension bigger than 1 in a hyperbolic manifold can be arbitrarily close to itself.

Here, we will discuss a tubular neighborhood theorem for an embedded complex codimension 1 submanifold in a complex hyperbolic n-manifold. We note that complex codimension 1 is real codimension 2. We provide an explicit estimate for this width that depends only on the codimension 1 volume of the embedded complex submanifold. We present two applications of this tubular neighborhood theorem: the first is a lower volume bound for such manifolds, and the second is an upper bound on the first eigenvalue of the Laplacian in terms of the geometry of the manifold.

A complex hyperbolic triangle group is the group generated by three complex reflections each fixing a complex line in complex hyperbolic space. Unlike for the case of constant curvature, the angles between the lines do not determine the triangle: there is one extra real parameter. In 2001 Schwartz gave a series of conjectures about when such groups are discrete. I will survey this landscape and report on recent progress.

Closed geodesics play a vital role in the understanding of hyperbolic surfaces and their moduli spaces, giving rise to a plethora of intertwined results and highlighting the many facets and depth of their geometry. While very rare, simple geodesics have played a role which has far exceeded their numbers, leading to spectacular results over the last 40 years. More recently, there has been a push to expand our understanding and extrapolate throughout the set of all closed geodesics.

This talk will be about some of these ventures, with a focus on counting problems.

We study the combinatorial interpretations of elements of Brin-Thompson group nV via block pair calculus and we proved that nV is torsion locally finite and contains infinitely many copies of additive group of rationals Q for n greater than or equal to 2. This is a joint work with Sadayoshi Kojima (arXiv:2603.18410).

We consider saddle connections on a translation surface in a hyperelliptic connected component of a stratum that do not intersect the interior of a distinguished saddle connection. For this restricted set of saddle connections, we show that it satisfies an L(log L)^d−2 growth rate, where d is the complex dimension of the hyperelliptic stratum. The upper bound holds for all translation surfaces in the hyperelliptic stratum while the lower bound holds for almost every surface in the hyperelliptic stratum. The proof of the lower bound uses horocycle renormalization. This is a joint work with Aulicino, Masur and Pan.

The boundary of complex hyperbolic plane carries two distinguished families of circles, arising as the ideal boundaries of its two types of maximal totally geodesic subspaces: complex geodesics, whose boundaries are called C-circles (or chains), and totally real Lagrangian planes, whose boundaries are called R-circles. In this talk, we illustrate the boundary geometry of complex hyperbolic triangle groups using these two families of circles. First, we study C-Fuchsian subgroups of some non-arithmetic complex hyperbolic lattices. We identify the real hyperbolic orbifolds they uniformize, and visualize the resulting chain orbits in Heisenberg coordinates. Second, we study a special family of complex hyperbolic triangle groups, investigating limit sets that appear to be built from configurations of R-circles.

Skein algebras are very important objects associated to the 3D quantum invariants. Roger and Yang introduced the punctured skein relations for the punctured surface, providing a natural quantization of the canonical Poisson structure on the decorated Teichmuller space. In this talk, we extend the canonical Poisson structure to the Fock–Goncharov higher decorated Teichmuller space associated with an arbitrary split semisimple algebraic group G with trivial center. For G=SL3 and G=SL4, we further construct an explicit quantization of these Poisson structures, yielding a natural generalization of Roger--Yang skein algebras. This is joint work in progress with Linhui Shen and Daping Weng.

In complex hyperbolic space, an elliptic isometry f is real if it preserves a Lagrangian plane. For such an f, the ideal boundaries of all preserved Lagrangian planes are pairwise disjoint and form a torus, which we call the fixed torus of f. In this talk, we first consider a real elliptic f of finite order n >= 3 and the Ford domain of < f >, arising as the limit of Dirichlet domains as the centre point approaches the boundary. The explicit cellular structure of the Ford domain will be shown, provided its centre lies on the fixed torus of f.

The boundary of the Ford domain is asymptotic to a complex hyperbolic cylinder—the lateral boundary of the union of all Cygan spheres centred on an R-circle, where the sphere centred at x has radius sqrt(1+x²). Using this cylinder as the geometric foundation of our method, we prove that a complex hyperbolic (n, ∞, ∞)-triangle group < I₁, I₂, I₃ > is discrete, faithful, and has the Ford domain with the same cellular structure as that of the R-Fuchsian group if and only if W_A is loxodromic and the isometric spheres I(W_B) and I(W_B⁻¹) are disjoint, where W_A = I₁ I₃ I₂ I₃ and W_B = I₁ I₂ I₃. This Ford domain is required to be centred at the unique common fixed point of I₁ and I₂.

Let X_p be the quotient of the closed disk obtained by identifying boundary points under rotation through angle 2π/p. For every 2 ≤ p ≤ 8, we construct a hyperbolic right-angled Coxeter group with nerve homeomorphic to X_p that admits a discrete, faithful, convex cocompact reflection representation into Isom(H⁵), whose limit set is homeomorphic to the index-p Pontryagin surface Π_p. Dimension five is optimal, since Π_p does not embed in S³. For p = 2, 3, the constructions are analytic and yield cyclically symmetric infinite families. This is joint work with Jiming Ma and Junseo Yoon.