Abstract:
In this talk, I will discuss the quasisymmetric classification of geometrically finite limit sets arising in conformal dynamics. The guiding question is: Given a topological type of limit set, which quasisymmetric equivalence classes can arise?

Two sharply different phenomena emerge.

• In the rigid regime, such as Sierpiński carpets, two geometrically finite limit sets are quasisymmetric if and only if their underlying dynamics are commensurable.
• In the universality regime, such as topological circles, any two geometrically finite limit sets are quasicircles, and hence are quasisymmetric.

I will discuss joint work with P. Haïssinsky that gives a complete answer for geometrically finite Kleinian groups. I will explain how the above dichotomy fits into a broader classification picture: the only obstructions to promoting a topological equivalence of limit sets to a quasisymmetric equivalence are carpet subsets and the coarse order structures at rank-two cut points. As an application, this allows us to obtain a quasi-isometric classification for finitely generated Kleinian group.

I will also describe the three main inputs in the proof: the decomposition of the group, the classification of topologically rigid groups, and the construction of piecewise linear models for Bowen--Series maps.

If time permits, I will discuss a broader conjectural framework that incorporates Julia sets of rational maps and other limit sets of conformal dynamical systems, together with some recent progress toward this conjecture.

Bio-Sketch:

Yusheng Luo is an Assistant Professor at Cornell University. He received his bachelor’s degree from the National University of Singapore in 2014 and his Ph.D. in mathematics from Harvard University in 2019. From 2019 to 2023, he held postdoctoral positions at the University of Michigan and Stony Brook University. His research lies in dynamics and geometry, especially complex dynamics and the corresponding moduli spaces, with connections to Kleinian groups, hyperbolic geometry, Teichmüller theory, and Berkovich dynamics.