Examples of non-ergodic actions on the SU(2)-character variety

par M. Fayssal Saadi

mardi 23 janv. 2024, 11:15 → 12:15 Europe/Paris

1R2-207

For a closed surface $\mathrm{\Sigma }$, the mapping class group $Mod(\mathrm{\Sigma })$ has a natural action by pre-composition on the SU(2)-character variety. When the surface is orientable, Goldman showed that the action is ergodic. Palesi on the other hand proved the ergodicity for non-orientable surfaces. 

In this talk, we consider a large subgroup $\mathrm{\Gamma }$ generated by Dehn-twists along a filling pair of multi-curves or a family of filling curves. Using a description based on square-tiled surfaces, we provide non-ergodic examples on the $SU(2)$-character variety of the non-orientable surface $N_4$ as well as examples on the representation variety of the orientable surface $S_2$, by showing that such a $\mathrm{\Gamma }$ admits explicit rational invariant functions.