Abstract: We will discuss the following result. For every nonarithmetic lattice Γ < SL2(ℂ) there is εΓ such that for every g ∈ SL2(ℂ) the intersection 

gΓg-1 ∩ SL2(ℝ) is either a lattice or a has critical exponent δ(gΓg-1 ∩ SL2(ℝ)) ≤ 1-εΓ. 

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.