Abstract

Consider a π₁-injective immersion f:S→Mfrom a compact surface S to a hyperbolic 3-manifold (M,h). Let Γ denote the copy of π₁(S) in Isom(H³) 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.