Agenda de l'ENS de Lyon

Dynamique sur les espaces de modules

ven 12 juil 2024



Amphi B, site Monod


Soutenance de SAADI Fayssal sous la codirection nationale de ZEGHIB Abdelghani et FUNAR Louis

Langue(s) des interventions
Description générale

In this thesis, we are interested in the dynamics of the mapping class subgroups on the SU(2)-character variety. More precisely, we deal with the ergodicity questions of a subgroup Γ of the mapping class group Modg,n of a compact surface Sg,n of genus g and n boundary components. These questions were naturally raised after Goldman’s proof of the ergodicity of mapping class groups on the SU(2)-character variety. The first general result in this direction is due to Funar and March´e by showing that the first Johnson subgroups act ergodically on the character variety, for any closed surfaces Sg. On the other hand, Brown showed the existence of an elliptic fixed point (or a double elliptic fixed point) for any subgroup generated by a pseudo Anosov element on the punctured torus S1,1. This led to the proof of the non-ergodicity of such subgroups by Forni, Goldman, Lawton, and Mateus by applying the KAM theory.

In the first part of the thesis, we study the natural dynamics of the moduli space of spherical triangles on S2 relating these dynamics to the dynamics of the mapping class group on the SU(2)-character variety of the punctured torus. The second part is devoted to the study of the existence of elliptic fixed for pseudo-Anosov homeomorphisms on the character varieties of punctured surfaces Sg,n, where g ∈ {0, 1}. By showing that near any relative character variety Xκ(π1(S1,1), SU(2)) of the once punctured torus, for an open dense set of levels κ, there exists a family of pseudo-Anosov elements that do not act ergodically on that level, in the case of the punctured torus S1,1. A similar result holds for a set of parameters B on XB(π1(S0,4), SU(2)), in the case of the four-punctured sphere S0,4. Then these results can be combined to construct a family of pseudo-Anosov elements on the twice-punctured torus S1,2 that admit an elliptic fixed point.

We discuss then the action of group T generated by Dehn-twist along a pair of filling multi-curves or along a family of filling curves on Sg. We show in this part that there exist two filling multi-curves on the surface of genus two S2 whose associated Dehn twists generate a group Γ acting non-ergodically on Hom(π1(S2), SU(2)) by finding explicit invariant rational functions. Similarly, We found invariant rational functions of a subgroup Γ generated by Dehn-twists along a family of filling loops on the character variety of the non-orientable surface N4.


Mots clés