Moduli of Riemann surface and Bers conjecture
Moduli of Riemann surface and Bers conjecture
This is a continuation of the October 24 talk. ÌýIt was Koebe who firstÌýproved that closed Riemann surface can be uniformized Ìýby Schottky groups. However Marden (1974) showed that not every Schottky group is generated by geometric circleÌýreflections in the complex plane, which is called "classical"(original definition by Schottky himself) Schottky group.Ìý Bers (1975)Ìýand Hejhal (1975) and AhlforsÌýmade detailed studies on Schottky space of moduli space of Riemann surface. And Bers made the following conjecture: "Every closed Riemann surface can be uniformized by classical Schottky group." ÌýIn this talk I will describe and present resolution of this conjecture based on two recent works. In fact, I will present the solution whichÌýactually answer a lot more to the original problem. First I will talk about smooth moduli space of Riemann surface, which we show that every closed Riemann surface is uniformizable by a Schottky group of Hausdorff dimension less than one. Second, I will give complete and sharpÌýclassification of Kleinian groups of Hausdorff dimension at most one. These two part works are independent and is based on completely different ideasÌýproofs. We prove the result on moduli space byÌýdeveloping ideas ofÌýCayley graph measure decompositions and norm of homological markings. The prove of the classification is based on application of deformation theory on local existenceÌýresult and rectifiability of invariant curves.Ìý