Critical exponent gap and leafwise dimension

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