Room 440, Astronomy-Mathematics Building, NTU

(台灣大學天文數學館 440室)

Entropy, Critical Exponent, and Immersed Surfaces in Hyperbolic 3-Manifolds

Lien-Yung Gao (University of Notre Dame &
University of Chicago)

Abstract

Consider a π_{1}-injective immersion *f:S**→M*from a compact surface *S* to a hyperbolic 3-manifold *(M,h)*. Let Γ denote the copy of π_{1}(*S*) in Isom(**H**^{3}) induced by the immersion. In this talk, I will discuss relations between two dynamics quantities: the critical exponent δ(Γ) 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.