Séminaire Géométrie et groupes discrets

Positivity and Representations of Surface Groups

by Prof. François Labourie (Université Côte d'Azur & IHES)

Amphithéâtre Léon Motchane (IHES)

Amphithéâtre Léon Motchane


Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette

Positivity is meant as a generalisation of the cyclic order on the circle. Associated to that is the notion of monotone maps from a cyclically ordered set in the circle.

Generalisations of the idea of positivity appeared to be crucial in understanding some connected components of the space of representations of a surface group in a Lie group G, although the common phenomenon was not figured out until recently.

In this talk, based on a preprint with Olivier Guichard and Anna Wienhard, I will start by examples generalizing this notion of cyclic order on the circle: convex curves or configurations in the plane, time like curve in Minkowski space. Then I will move to the general geometry of parabolic spaces and explain why the notion of positivity relates to special configurations of pairwise transverse triples and quadruples of points. This notion of positivity, which abides simple combinatorial properties, allows to define positive — or monotone — curves, then positive representations of surface groups.

I will then sketch the proof of our main result: positive representations are Anosov and fill up connected components of the space of representations.


IHES Covid-19 regulations:

- all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
- speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
- Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
- Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.


Organized by

Fanny Kassel