Appears in collection : 2024 - T2 - WS2 - Group actions with hyperbolicity and measure rigidity

We will discuss the following result. For every nonarithmetic lattice $\Gamma < \mathrm{SL}_2(\mathbb{C})$ there is $\varepsilon \Gamma$ such that for every $g \in \mathrm{SL}_2(\mathbb{C})$ the intersection $g\Gamma g^{-1} \cap \mathrm{SL}_2(\mathbb{R})$ is either a lattice or a has critical exponent $\delta(g\Gamma g^{-1} \cap \mathrm{SL}_2(\mathbb{R})) \leq 1-\varepsilon \Gamma$. This result extends Mohammadi-Margulis and Bader-Fisher-Milier-Strover. We will focus on an ergodic component of the proof, asserting certain preservation of entropy-contribution under limits of measures.