Positivity and Surface Groups / Ghost algebra, Poisson bracket and convexity (François LABOURIE, Nice)

19.02.2025 10:00 – 12:00

Positivity and surface groups

Positivity in a generalizsd flag manifold $M$ was introduced by Guichard and Wienhard. It generalizes both Lusztig poitivity and the geometry of the Shilov boundary. We will describe this notion which is based on a simple « nesting » property and show how it helps defining positivity maps of the circle in $M$, as well as representation of surface groups with properties mimicking those arising as monodromies of hyperbolic structures. Based on work with Guichard, Wienhard, and then Pozzetti and Beyrer.

Ghost algebra, Poisson bracket and convexity

The moduli space of Anosov representations of a surface group in a semisimple group – an open set in the character variety – admits many more natural functions than the regular functions. We will study in particular length functions and correlation functions. Our main result is a formula that computes the Poisson bracket of those functions using some combinatorial devices called ghost polygons and ghost bracket encoded in a formal algebra called the ghost algebra related in some cases to the swapping algebra. As a consequence of our main theorem, we show that the set of those functions – length and correlation – is stable under the Poisson bracket. We give two applications: firstly in the presence of positivity we prove the convexity of length functions, generalizing a result of Kerckhoff in Teichmüller space. Based on work with Bridgeman.

Lieu

Bâtiment: Conseil Général 7-9

Room 1-07, Séminaire "Groupes de Lie et espaces de modules"

entrée libre