Schedule, Notes/Handouts & Videos

We first introduce a class of metric-like functions on hyperbolic groups, called hyperbolic metric potentials. This is a class of functions general enough to include word-metrics, quasi-morphisms, and the fundamental weights of Anosov representations. Then, given a tuple (f1,...,fd) of such functions, we introduce the notion of translation cone, an analogue of the limit cone introduced in the setting of linear algebraic groups by Benoist in the 90s. We establish analogues of Benoist's results as well as additional hyperbolic features on this cone. We then turn to a more precise asymptotic analysis: counting. We introduce the analogue of the growth indicator function, introduced in early 2000s by Quint again in the linear setting. We show that this function is always strictly concave and C1, which generalize results of Quint, Sambarino, Kim-Oh-Wang, etc. Finally, we relate this function to a multi-dimensional generalization of the Manhattan curve, which is a curve of Poincaré exponents, introduced in dimension 2 by Burger in the 90s, and recently studied in our more general setup by Tanaka, and Cantrell--Tanaka. Joint work with Stephen Cantrell and Eduardo Reyes.

In 2008, Guichard-Wienhard showed that PSL(4,R)-Hitchin representations of surface groups are exactly the holonomies of "properly convex foliated" projective structures on unit tangent bundles of surfaces. In this talk we will discuss phenomena that emerge in this story, with an eye towards connections to other directions in projective geometry and higher Teichmüller theory. After presenting the setting, we will explain two such connections. First, the leaves of a codimension-one foliation show a new point-set topological property of the (non-Hausdorff) space of projective equivalence classes of convex domains in RP(n), which in particular answers an old question of Benzécri. Second, we describe how a codimension-two foliation is connected to a concrete and general construction that reproduces some flows that are studied on the dynamical side of higher Teichmüller theory. Part of this is based on joint work with Max Riestenberg.

A translation surface is a closed surface obtained by gluing edges of a polygon by translations. The group GL_2(R) acts on the collection translation surfaces of a fixed genus g. Eskin and Mirzakhani classified probability measures that are invariant under SL_2(R) and, more generally, under the upper triangular subgroup. In the talk we will discuss a new extension that describes probability measures invariant under the horocyclic flow, conjectured by Forni. We also present an application to billiards with rational angles.

In the exceptional real split Lie group $G_2'$,  Collier and Toulisse introduced cyclic Higgs bundles which have the property of defining representations admitting equivariant holomorphic curves on the pseudosphere. The picture they depict is very similar to that for other families of representations previously constructed in rank 2 Lie groups, which are Anosov and admit associated geometric structures.
In a joint work with Parker Evans we study two families of such representations in $G_2'$. I will  describe these families and present how we construct geometric structures modelled on the $G_2'$ flag manifolds whose holonomies are these representations, using a unifying construction that allows us to reinterpret previously known constructions in other rank 2 Lie groups. I will also present our more recent results about the Anosov properties of these representations.