Adjoining a Haar-Generic Matrix to a Parabolic-Free Subgroup of SL₂(Qₚ)

Let Γ be a countable subgroup of SL₂(ℚ_p) containing no noncentral element of trace 2 or −2. We prove that, outside a Haar-null set of g ∈ SL₂(ℚ_p), the generated group ⟨Γ,g⟩ has the same property. Thus adjoining one random element preserves the absence of parabolics almost surely, answering Question 4 in Miklós Abért's 2010 list. The key elementary lemma classifies the problematic one-variable generalized words: if all constants lie in a parabolic-free subgroup of SL₂ and the trace of the word is identically 2 or −2, then the word map is identically I or −I. The proof evaluates the word on I+tN over the nilpotent cone. Its two highest coefficients force a central endpoint product and cancellation of the outer exponents, reducing the word by conjugation and induction. This manuscript is unrefereed and makes no absolute priority claim. Research status: Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-use disclosure: AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text.