Densities in free groups and ${\mathbb{Z}}^k$, Visible Points and Test Elements

Ilya Kapovich, Igor Rivin, Paul Schupp +1

In this article we relate two different densities. Let $F_k$ be the free group of finite rank $k\ge 2$ and let $\alpha$ be the abelianization map from $F_k$ onto ${\mathbb{Z}}^k$. We pr…