|Location:||MSRI: Simons Auditorium|
It was Koebe who first proved that closed Riemann surface can be uniformized by Schottky groups. However Marden showed that not every Schottky group is generated by geometric circle reflections in the complex plane, which is called "classical"(original definition by Schottky himself) Schottky group. Bers and Hejhal and Ahlfors made 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 a classical Schottky group." In this talk I will describe and present resolution of this conjecture based on two recent works. In fact, I will present the solution which actually 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 sharp classification of Kleinian groups of Hausdorff dimension at most one. These two part works are independent and is based on completely different ideas proofs. We prove the result on moduli space by developing ideas of Cayley graph measure decompositions and norm of homological markings. The prove of the classification is based on application of deformation theory on local existence result and rectifiability of invariant curves.