Strongly dense free subgroups of semisimple algebraic groups II

TL;DR

This paper extends the existence of strongly dense free subgroups in semisimple algebraic groups to broader fields, and explores related questions for surface groups, highlighting new conditions based on field transcendence.

Contribution

It generalizes previous results by establishing the presence of strongly dense free subgroups over fields with certain transcendence degrees in all characteristics.

Findings

Strongly dense free subgroups exist over fields with transcendence degree ≥1 in characteristic zero.

Such subgroups exist over fields with transcendence degree ≥2 in positive characteristic.

The paper also investigates related properties for surface groups.

Abstract

It was shown in Part I that there exist strongly dense free subgroups in any semisimple algebraic group over a large enough field. These are nonabelian free subgroups all of whose subgroups are either cyclic or Zariski-dense. Here we show that the same is true for as long as the transcendence degree of the field is at least $1$ in characteristic zero and at least $2$ in positive characteristic. We also consider related questions for surface groups.