For a (finitely generated) subgroup of a (finitely generated) group, we will consider two properties of this subgroup: the separability of this group and its subgroup distortion. The separability of a subgroup measures whether the property of an element not lying in this subgroup is visible by taking some finite quotient. We will give a characterization on whether a subgroup of a 3-manifold group is separable. The subgroup distortion compares the intrinsic and extrinsic geometries of a subgroup. For an arbitrary subgroup of a 3-manifold group, we prove that the subgroup distortion can only be linear, quadratic, exponential and double exponential. It turns out that these two properties (subgroup separability and subgroup distortion) are closed related for subgroups of 3-manifold groups. The subgroup distortion part is joint work with Hoang Nguyen.

In 1908, Paul Koebe conjectured that every open connected set in the plane is conformally diffeomorphic to an open connected set whose boundary components are either round circles or points. The Weyl problem, in the hyperbolic setting, asks for isometric embedding of surfaces of curvature at least -1 into the hyperbolic 3-space. We show that there are close relationships among the Koebe conjecture, the Weyl problem and the work of Alexandrov and Thurston on convex surfaces. This is a joint work with Feng Luo.

TBA