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

Abstract

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