Coarse Extrinsic Geometry and surface-by-free groups

JOINT WITH SIMONS SEMESTER

The notion of distortion of finitely generated subgroups H of finitely generated groups G, introduced by Gromov, is a first step towards studying extrinsic geometry of subgroups in the coarse setup. When the group and subgroup are hyperbolic, finer information can be obtained in the presence of a Cannon-Thurston map. We shall discuss applications towards proving quasiconvexity results. When the ambient group G is surface-by-free, hyperbolic, such quasiconvexity results are an ingredient leading to a cubulation result in certain situations. This will be an expository talk drawing from joint work with several people including Rafi, Manning and Sageev.