Overview
This research investigates the concept of redundancy within homomorphisms from free groups to the topological group $\operatorname{PSL}_2\left(\mathbb{Q}_p\right)$. The study focuses on the action of $\operatorname{Aut}\left(F_n\right)$ on the set of epimorphisms $\operatorname{Epi}\left(F_n,G\right)$, where $F_n$ denotes a free group on $n$ generators and $G$ is a topological group.
Research Context
The definition of a redundant homomorphism is central to this work. A homomorphism $f \in \operatorname{Epi}\left(F_n,G\right)$ is defined as redundant if there exists a proper free factor $A$ of $F_n$ such that the image of $A$ under $f$, denoted $f(A)$, is dense in $G$. The overarching context involves understanding the generation properties of topological groups when acted upon by automorphisms of free groups.
Approach
The research examines specific conditions for redundancy. It considers the case where $G$ is precisely $\operatorname{PSL}_2\left(\mathbb{Q}_p\right)$, and the number of generators $n$ for the free group $F_n$ is at least 3 ($n \ge 3$). Within this setup, two primary scenarios for the image of the homomorphism $f$ were analyzed: one where the image is torsion-free, and another where it is not.
Findings
- For $n \ge 3$ and $G = \operatorname{PSL}_2\left(\mathbb{Q}_p\right)$, every homomorphism $f \in \operatorname{Epi}\left(F_n,G\right)$ with a torsion-free image is redundant.
- Sufficient conditions were identified that lead to $f$ being redundant, even in instances where its image is not torsion-free.
Why This Matters
The study clarifies fundamental properties of how free groups map onto specific topological groups, particularly regarding redundant generators. These findings contribute to the understanding of group generation and the structure of homomorphisms under certain conditions.