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Topology/Geometry Seminar

Surface Subgroups in Cocompact Kleinian Groups

Zhenghao Rao (Brown)

Location:  Hill 705
Date & time: Tuesday, 26 September 2023 at 4:00PM - 5:00PM

Kahn and Markovic proved the Surface Subgroup conjecture for closed hyperbolic 3-manifolds more than ten years ago. The surface subgroup they constructed can be as close as possible to Fuchsian. However, a closed hyperbolic 3-manifold can also have surface subgroups far away from being Fuchsian. Actually, provided any genus-2 quasi-Fuchsian group ? and cocompact Kleinian group G, then for any K>1, we can find a surface subgroup H of G that is K-quasiconformally conjugate to a finite index subgroup F<?. We will point out the difference between my theorem and the original Surface Subgroup Theorem, discuss the proof idea, and introduce some applications. For instance, we can use this theorem to prove that the set of Hausdorff dimensions of limit sets of surface subgroups of G is dense in [1,2].

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