ResearchHere are some slightly more specific (and more technical) blurbs describing things I am currently looking at:
Anosov representations have come to be considered a good generalization of convex cocompactness in the higher-rank setting. Is there a corresponding notion of geometric finiteness?
Teichmüller space has many useful metrics on it. What about higher Teichmüller spaces?
In the case of the Hitchin component Bridgeman-Canary-Labourie-Sambarino defined a pressure metric, using dynamical invariants associated to the geodesic flow on the group, and there may be a complex-analytic interpretation of this metric, by work-in-progress of Deroin-Tholozan.
Here is a different idea: can we define an analogue, for higher Teichmüller spaces, of Thurston's (asymmetric) Lipschitz metric on Teichmüller space?
Is CAT(0) the "natural" / "best" notion of NPC in the context of groups?
Under this fuzzy umbrella comes tricky questions like "Are all Gromov-hyperbolic groups CAT(0)?", the Brady-Bestvina wager (True or false: the fundamental group of a NPC manifold is Gromov-hyperbolic iff it contains no higher-rank free abelian subgroup), and so on.
Where is the boundary between CAT(0) and cubulated?