Convergence of quasifuchsian hyperbolic 3-manifolds
Dynamics on Moduli Spaces April 13, 2015 - April 17, 2015
Location: MSRI: Simons Auditorium
20H10 - Fuchsian groups and their generalizations [See also 11F06, 22E40, 30F35, 32Nxx]
30F35 - Fuchsian groups and automorphic functions [See also 11Fxx, 20H10, 22E40, 32Gxx, 32Nxx]
37-XX - Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX]
37F30 - Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems
Thurston's Double Limit Theorem provided a criterion ensuring convergence, up to subsequence, of a sequence of quasifuchsian representations. This criterion was the key step in his proof that 3-manifolds which fiber over the circle are geometrizable. In this talk, we describe a complete characterization of when a sequence of quasifuchsian representations has a convergent subsequence. Moreover, we will see that the asymptotic behavior of the conformal structures determines the ending laminations and parabolic loci of the algebraic limit and how the algebraic limit ``wraps'' inside the geometric limit. (The results described are joint work with Jeff Brock, Ken Bromberg, Cyril Lecuire and Yair Minsky.)
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