The Refined Toledo Invariant of a Non-Archimedean Surface Group Representation

by Marc Burger (ETH Zürich)

Monday Jun 8, 2026, 2:00 PM → 3:15 PM Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Let S be a compact oriented surface with boundary, Γ its fundamental group, and G a simple algebraic group defined over Q such that the symmetric space associated to G(R) is Hermitian of tube type. Given a real closed field F and a canonical presentation of Γ, we define for a representation of Γ in G(F) an invariant taking values in an ordered abelian group A(F), called the refined Toledo invariant, as it generalizes the Toledo number in the case F = R. The group A(F) has a geometric interpretation as the group of signed areas for polygons in the Hilbert geometry associated to the upper half plane over F. The goal of the talk is to describe the construction of this invariant and to explain how it solves the problem of characterizing points in the real spectrum compactification of the space of maximal representations of Γ into G(R).