A strong height gap theorem for $PGL_2$

Abstract: The height gap theorem states that the finite subsets $F$ of matrices generating non-virtually solvable groups have normalized height $\widehat{h}(F)$ bounded below by a constant. It was first proved by Breuillard and another proof was given later by Chen, Hurtado and Lee. In this paper we show that when the set $F$ is contained in a maximal arithmetic subgroup $\Gamma$ of $G = PGL_2(\mathbb{R})^a \times PGL_2(\mathbb{C})^b$, $a+b \ge 1$, the height bound for the case when $F$ generates a Zariski dense subgroup of $G$ over $\mathbb{R}$ is proportional to $\log(covol(\Gamma))$, the function of the covolume of $\Gamma$. This result strengthens the theorem for the lattices of large covolume and has various applications including a strong version of the arithmetic Margulis lemma for $PGL_2(\mathbb{R})^a \times PGL_2(\mathbb{C})^b$.