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

Let \(\Gamma\) be a countable subgroup of \(\G(\mathbb Q_p)\) containing no noncentral element of trace \(2\) or \(-2\) . We prove that, outside a Haar-null set of \(g\in\G(\mathbb Q_p)\) , the generated group \(\langle\Gamma,g\rangle\) 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 \(\G\) 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.