A strong height gap theorem for $PG{L}_2$

Abstract

The height gap theorem states that the finite subsets $F$ of matrices generating non-virtually solvable groups have normalized height $\hat{h}(F)$ 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 $\hat{I}“$ of $G=PG{L}_2(\mathbb{R}{)}^a\times PG{L}_2(\mathbb{C}{)}^b$, $a+b\ge 1$, the height bound for the case when $F$ generates a Zariski dense subgroup of $G$ over $\mathbb{R}$ is proportional to $\log (covol(\hat{I}“))$, the function of the covolume of $\hat{I}“$. 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 $PG{L}_2(\mathbb{R}{)}^a\times PG{L}_2(\mathbb{C}{)}^b$.

16 pages