Critical exponent gap and leafwise dimension

We show that for every nonarithmetic lattice $\Gamma<{\rm SL}_2(\mathbb{C})$ there is a gap $\varepsilon_\Gamma>0$ such that for every $g\in {\rm SL}_2(\mathbb{C})$ the intersection ${\rm SL}_2(\mathbb{R})\cap g\Gamma g^{-1}$ is either a lattice in ${\rm SL}_2(\mathbb{R})$ or has critical exponent $\delta({\rm SL}_2(\mathbb{R})\cap g\Gamma g^{-1}) \leq 1 - \varepsilon_\Gamma$.