4:00 pm Monday, April 5, 2010
Colloquium: Limit points of limit points of limit points... volume spectra, length spectra and limit ordinals in hyperbolic geometry by Jeff Brock (Brown) in RLM 6.104
A homeomorphism f of a surface S of negative Euler-characteristic has a dual role in low dimensional geometry and topology: it determines a loop on the moduli space of Riemann surfaces, and it naturally gives rise to a 3-manifold, the `mapping torus' M(f) of f. When f is "pseudo-Anosov" each of these constructions determines an invariant of the isotopy class of f: the Weil-Petersson geodesic length of the loop, on the one hand, and the hyperbolic volume of M(f), on the other. In this talk I'll discuss the surprising fact that these quantities are in bounded ratio, and how this motivates the result that the length spectrum for moduli space is well-ordered with order type \omega^\omega. We'll focus on the case of the modular surface, the upper-half-plane modulo SL(2,Z), and how such connections are intimately related in this case to continued fraction expansions associated to the fixed points at infinity for f. I'll conclude with a discussion of how, despite these and other connections between hyperbolic volume and Weil-Petersson length, there can be no exact formula relating the two, answering in the negative a conjecture of Manin and Marcolli. This talk represents joint work with Howard Masur, Yair Minsky and Juan Souto. Submitted by
|
|