Simple Torsion-Free Linear Groups from Global Division Algebras

We investigate the reduced norm-one group $\mathrm{SL}_1(D)$ of a central division algebra $D$ of odd degree over $\mathbb{Q}$. Using the Brauer-Hasse-Noether theorem and the Margulis-Platonov theorem, we derive a criterion for abstract simplicity in terms of local division completions. For a division algebra of degree $15$ with local invariants $1/3$, $-1/3$, $1/5$, and $-1/5$, the construction yields a candidate example of an infinite, torsion-free, nonabelian simple linear group. This would provide an affirmative answer to Problem 16.78 of the Kourovka Notebook. The argument was generated using an AI system and has not yet been independently verified by human mathematicians.