Some Questions — Question 4

Definitions and mathematical statement

Let $p$ be a prime, let ${\mathbb{Q}}_p$ be the field of $p$-adic numbers, and set \[ G=\operatorname{SL}_2({\mathbb{Q}}_p)=\{A\in M_2({\mathbb{Q}}_p):\det A=1\}. \] A possible intended matrix convention calls $A\in G$ parabolic when $\operatorname{tr}(A)\in\{2,-2\}$ and $A\notin\{I,-I\}$, equivalently when it is a noncentral matrix with a repeated eigenvalue. The record does not define "parabolic"; this convention must be checked, particularly because terminology for actions on the Bruhat--Tits tree can differ.

Let $\Gamma\leq G$ be countable and contain no parabolic elements. For $g\in G$, $\langle\Gamma,g\rangle$ denotes the smallest subgroup containing $\Gamma$ and $g$. The source asks whether adjoining a random $g$ almost surely preserves the absence of parabolics. Because $G$ is noncompact, Haar measure on $G$ is not itself a probability distribution. The accompanying research summary makes the randomness precise as follows: let $m_G$ be Haar measure and let $\mu$ be any Borel probability measure satisfying $\mu\ll m_G$. Show that \[ \mu\left(\{g\in G:\langle\Gamma,g\rangle \text{ contains no parabolic element}\}\right)=1. \] Under the matrix convention above, the event can be written explicitly as \[ \forall A\in\langle\Gamma,g\rangle, \quad\operatorname{tr}(A)\in\{2,-2\}\ \Longrightarrow\ A\in\{I,-I\}. \] No discreteness assumption on $\Gamma$ is supplied.

The registry marks this solved by its own contributor. Its summary claims a proof for every such absolutely continuous probability measure. It describes a purported new algebraic lemma about generalized word maps on $\operatorname{SL}_2$ over algebraically closed fields of characteristic zero: identically $\pm2$ trace is said to force formal collapse to the constant $\pm I$. The summary says the no-parabolics assumption forces an endpoint product $a_0\cdots a_k=\pm I$, and that induction on the number of variable factors uses the top two coefficients of $\operatorname{tr}(w(I+tN))$ over the nilpotent cone. No proof or precise definition of formal collapse is supplied. These are attributed proof claims, with novelty qualified by the contributor's own verification; they are not certified results in this draft.

Short English statement

Given a countable subgroup of $\operatorname{SL}_2({\mathbb{Q}}_p)$ with no parabolic elements, must adjoining an independently chosen element with any Haar-absolutely-continuous probability law almost surely preserve that property? The summary specifies this probability convention and claims an unpublished proof; the meaning of parabolic and that proof require review.

Sources

• Local UnsolvedMath registry, permanent ID 1200004, source code catalogue entry, "Some Questions --- Question 4"; supplied status solved, classification SOLVED-BY-YOU, updated 2026-08-06. The proof and novelty claims are those of the supplied research summary.
• Abert, Some questions, author list dated 2010, Question 4, https://www.renyi.hu/~abert/questions.pdf. The record reports manual transcription and access on 2026-07-29; it says that the 2010 list was inspected and current status was not independently verified. Status and rights: NEEDS_REVIEW; public author-hosted PDF.
• No separate publication, proof manuscript, or external citation for the claimed algebraic lemma or claimed solution is supplied.