Putting a hyperbolic metric on a complete finite-type surface gives us a linear representation (the holonomy representation) with many nice geometric and dynamical properties: for instance it is discrete and faithful, and in fact stably quasi-isometrically embedded, and the group acts on its limit set with north-south dynamics. This picture can be generalised in (at least) two ways. First, the notion of geometric finiteness generalises this picture in the context of rank-one Lie groups such as PSL(2,R) or PSL(2,C). Second, Anosov representations generalise this picture to higher-rank Lie groups such as PSL(d,K) for d>2.
In the first talk on Monday, I will introduce relatively Anosov representations as a common generalisation of Anosov representations on the one hand and geometric finiteness in rank one on the other. I will mention projectively visible subgroups as examples, and also discuss various variations on the notion.
In the second talk Tuesday morning, I will briefly discuss some aspects of the proofs. The general theme here will be how the lack of compactness makes things trickier in the relative case, and some ways around this.
This generalises work of Canary–Zhang–Zimmer and is mostly joint work with Andrew Zimmer.