Critical exponent gap and leafwise dimension

Abstract

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

arXiv comment49 pages