2025 2K-GATE One day Geometric Topology Festival 

Venue

KIAS(Korea Instiute for Advanced Study), Room 1503. Direction Link

Registration Link (due August 25) : here

Speakers

Yair Minsky (Yale University)

Hidetoshi Masai (Musashino Art University)

Khanh Le (Rice University)

Seung-Yeon Ryoo (Caltech)

Schedule

9:00-9:30 Coffee 

9:30-10:30 Yair Minsky

   Title: Horocycles and laminations in abelian covers of hyperbolic surfaces 

   Abstract: We consider topological dynamics of horospherical flow in hyperbolic manifolds.  Since the work of Hedlund in the 1930's it is known, for example, that horocycles in closed hyperbolic surfaces are always dense and equidistributed. In infinite-volume manifolds there is much more flexibility. The case of Z^d-covers of a closed hyperbolic manifold M has a particularly nice blend of flexibility and rigidity. There is an interesting connection between closures of horospherical orbits and optimal equivariant 1-Lipschitz functions. Such functions achieve their maximal stretch on the lift of a geodesic lamination in M, and the geometry of this lamination influences the behavior of the orbit closures.  We can give a complete description of horocycle orbit closures in the case of Z-covers of a closed hyperbolic surface. In particular we were surprised to find that all proper orbit closures are fractal in some sense, and yet have integer Hausdorff dimension. This is joint work with James Farre and Or Landesberg.

10:40-11:40 Hidetoshi Masai

   Title: On the extremal length of the hyperbolic metric

   Abstract: The extremal length of curves can be viewed as a notion of length determined by the Riemann surface structure. Mart\'inez-Granado and Thurston proved that the square root of the extremal length function extends continuously to the space of geodesic currents. This allows us to consider the extremal length of the Liouville current associated with the hyperbolic structure of the surface. In this talk, I will discuss the extremal length of Liouville currents, together with related extremal metrics.

11:40-13:00 Lunch 

13:00-14:00 Khanh Le 

   Title: Biorderability and free-by-cyclic groups

   Abstract: A group is bi-orderable if it admits a total ordering that is left and right invariant. Orderable groups have received recent attention due to their connection with dynamical group theory and with 3-manifold groups via the L-space conjecture. In this talk, I will describe various criterions for a free-by-cyclic group to be bi-orderable. If time permits, I will discuss some computational experiment exploring the genericity of these criterions. This is a joint work with Jonathan Johnson.

14:00-14:30 Coffee 

14:30-15:30 Seung-Yeon Ryoo

   Title: Asymptotics of Riemannian Lie groups with nilpotency step 2

   Abstract: We compare asymptotic Riemannian and sub-Riemannian metrics in step-2 nilpotent Lie groups. More precisely, given a (sub-)Riemannian metric on a step-2 nilpotent Lie group, we show that there exists a Carnot group metric whose square remains at a bounded distance from the square of the original metric; this is done by a new perturbation technique for sub-Riemannian geodesics, which allows control on the vertical coordinate of the endpoint. As a consequence, we obtain a refined estimate on the error term in the asymptotic expansion of the volume of metric balls, and we also obtain a finer estimate on the rate of convergence to the asymptotic cone. This estimate is not true for nilpotency step 3 or for Finsler metrics. This is a joint work with Enrico Le Donne, Luca Nalon, and Sebastiano Nicolussi Golo. 

Organizers

Harry Hyungryul Baik (KAIST)

Inhyeok Choi (KIAS)

Sang-hyun Kim (KIAS)