Redundant generation in ${\mathrm{PSL}}_2\left({\mathbb{Q}}_p\right)$

Abstract

Given a free group on $n$ generators, $F_n$, and a topological group $G$, $\mathrm{Aut}\left(F_n\right)$ acts on $\mathrm{Epi}\left(F_n,G\right)$, the set of homomorphisms $f:F_n\to G$ with dense image. $f\in \mathrm{Epi}\left(F_n,G\right)$ is called redundant if there is a proper free factor $A<F_n$ whose image under $f$ is dense in $G$. For $n\ge 3$, $G={\mathrm{PSL}}_2\left({\mathbb{Q}}_p\right)$, we prove that every $f\in \mathrm{Epi}\left(F_n,G\right)$ with torsion-free image is redundant, and give sufficient conditions for $f$ to be redundant even when its image is not torsion-free.

9 pages