Rigidity, deformations and limits of maximal representations

• Applicant: Professorin Dr. Anna Wienhard, since 8/2024
• Subject Area: Mathematics
• Term: since 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 338644254

Lattices, i.e. discrete subgroups of finite covolume, in higher rank Lie groups are well understood by Margulis celebrated arithmeticity theorem. They all come from number theoretic constructions. Surprisingly little is known about discrete subgroups of infinite covolume: in general no algorithm can be given to check if finitely many elements of a semisimple Lie group generate a discrete subgroup and there is no general procedure to verify if a representation of a discrete group is injective. In this perspective, the existence of higher Teichmüller spaces, which are components in the representation varieties of the fundamental group of a surface in various Lie groups of higher rank consisting only of discrete and injective representations is quite surprising, and the discoveryof Anosov representations, which is a rich class of discrete subgroups with good geometric and dynamical properties has been a major breakthrough.The study of such subgroups forms a very rich and promising field: it allows for a combination of geometric ideas and constructions inspired from low dimensional topology, mixed with the more rigid and rich structure pertinent to higher rank. Maximal representations give one family of higher Teichmüller spaces, and important examples of Anosov representations.The priority program "Geometry at infinity" (SPP 2026) aims at understanding convergence and limits in geometric-topological settings, and asymptotic properties of objects of infinite size. We believe that the non-compact space of maximal representations form an ideal setting in which to implement this philosophy. In our project we continue embodying the perspective "geometry at infinity" in the study of maximal representations: we investigate important geometrical and dynamical asymptotic properties of maximal representations, we develop a better understanding of points at infinity in various compactifications of the space of maximal representations, and we describe new ways of moving in their parameter space.This is a proposal for a project renewal. We implement the three cross-sectional topics at the basis of the priority program to study geometric properties of discrete subgroups of semisimple Lie groups, in particular those arising as images of maximal representations and Anosov representations. We pursue the following three research directions:1) discover new relations between asymptotic properties of Anosov representations, such as the entropy or the orbit growth in pseudo-Riemannian symmetric spaces, and the Hausdorff dimension of naturally defined subsets of the limit sets.(2) define new ways to move in the space of maximal representations via earthquakes, cataclysms and generalized twist flows, and study convergence of geometric quantities along these deformations.(3) study compactifications of maximal representation as a more combinatorial object and discover the geometry of new higher rank features.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity
• Ehemalige Antragstellerin: Professorin Dr. Maria Beatrice Pozzetti, until 8/2024