A strong height gap theorem for $PGL_2$

TL;DR

This paper establishes a height gap theorem for subsets of $PGL_2$ within maximal arithmetic subgroups, linking the height bound to the covolume of the lattice, with implications for the arithmetic Margulis lemma.

Contribution

It extends the height gap theorem to maximal arithmetic subgroups of $PGL_2$, relating height bounds to covolume and strengthening results for large covolume lattices.

Findings

Height bound proportional to log of covolume of the lattice

Strengthens the theorem for lattices with large covolume

Applications include a strong version of the arithmetic Margulis lemma

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 $PGL_2(\mathbb{R})^a \times PGL_2(\mathbb{C})^b$.