Unit equations on quaternions

Abstract

A classical result about unit equations says that if $\Gamma_1$ and $\Gamma_2$ are finitely generated subgroups of $\mathbb C^\times$, then the equation $x+y=1$ has only finitely many solutions with $x\in\Gamma_1$ and $y\in \Gamma_2$. We study a noncommutative analogue of the result, where $\Gamma_1,\Gamma_2$ are finitely generated subsemigroups of the multiplicative group of a quaternion algebra. We prove an analogous conclusion when both semigroups are generated by algebraic quaternions with norms greater than 1 and one of the semigroups is commutative. As an application in dynamics, we prove that if $f$ and $g$ are endomorphisms of a curve $C$ of genus $1$ over an algebraically closed field $k$, and $\mathrm{deg}(f), \mathrm{deg}(g)\geq 2$, then $f$ and $g$ have a common iterate if and only if some forward orbit of $f$ on $C(k)$ has infinite intersection with an orbit of $g$.