Free subgroups of special linear groups

Abstract

We present a proof of the following claim. Suppose that $n$ is an integer such that $n>1$ and that $k$ is any field. Suppose that $g$ is an element of $\mathrm{SL}(n,k)$ of infinite order. Then the set $\{h\in\mathrm{SL}(n,k)\mid <g,h>$ is a free group of rank two$\}$ is a Zariski dense subset of $\mathrm{SL}(n,\bar{k})$ where $\bar{k}$ is an algebraic closure of $k$.