Title: Entropy, Critical Exponent, and Immersed Surfaces in Hyperbolic 3-Manifolds
Speaker: Lien-Yung Kao (University of Notre Dame)
Abstract: Consider a $\pi_{1}$--injective immersion $f:S\to M$ from a compact surface $S$ to a hyperbolic 3--manifold $(M,h)$. Let $\Gamma$ denote the copy of $\pi_{1}S$ in $\mathrm{Isom}(\mathbb{H}^{3})$ induced by the immersion. In this talk, I will discuss relations between two dynamics quantities: the critical exponent $\delta(\Gamma)$ and the topological entropy $h_{top}(S)$ of the geodesic flow for the immersed surface $(S,f^{*}h)$. These dynamics relations lead us to geometry results: through these relations one can characterize certain hyperbolic 3--manifolds such as Fuchsian manifolds, quasi-Fuchsian manifolds, and almost-Fuchsian manifolds. If time permits, I will also discuss applications of these relations to the moduli space of $S$ introduced by C. Taubes.
Seminar URL: https://research.math.osu.edu/topology/
