Discrete subgroups of semisimple Lie groups beyond Anosov

• Applicant: Professorin Dr. Maria Beatrice Pozzetti
• Subject Area: Mathematics
• Term: since 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 427903332

Discrete subgroups of semisimple Lie groups can be understood as fundamental groups of locally symmetric spaces, manifolds modelled on particularly rich geometries.Surprisingly little is known about such groups, and only recently a robust criterion for stability was given: Anosov subgroups are strongly undistorted subgroups, a property that persists under small deformations. In most interesting cases, such subgroups admit rich deformation spaces. The drawback of this definition is that it can only be satisfied by hyperbolic groups, which in particular cannot contain any higher rank abelian subgroup, and thus cannot explore much of the flat geometry of a symmetric space. Through different concrete subprojects I aim at going beyond the notion of Anosov subgroups, and thus enlarge our understanding of discrete subgroups of semisimple Lie groups in three different directions: I want to understand representations in the boundary of Anosov representations, both generalizing ideas from low dimensional topology, and discussing new cohomological notions of geometric infiniteness, I plan to develop a weaker stability condition that also includes algebraically richer groups, thus providing a notion of stable actions that includes groups with non-trivial abelian subgroups, and I plan to further develop the study of Anosov (and maximal) actions on infinite dimensional symmetric spaces, as I believe that this can bring new insight on an interesting class of actions on Hilbert spaces.

• DFG Programme: Independent Junior Research Groups