We show that the first Johnson subgroup of the mapping class group of a surface $\Sigma$ of genus greater than $1$ acts ergodically on the moduli space of representations of $\pi_1(\Sigma)$ in $SU_2$. Our proof relies on a local description of the latter space around the trivial representation and on the Taylor expansion of trace functions.
"The first Johnson subgroups act ergodically on $SU_2$-character varieties." J. Differential Geom. 95 (3) 407 - 418, November 2013. https://doi.org/10.4310/jdg/1381931734