A composition law and refined notions of convergence for periodic continued fractions

We define an equivalence relation on periodic continued fractions with partial quotients in a ring $\mathcal{O}\subseteq \mathbf{C}$, a group law on these equivalence classes, and a map from these equivalence classes to matrices in ${\mathrm{G}\mathrm{L}}_2(\mathcal{O})$ with determinant $\pm 1$. We prove this group of equivalence classes is isomorphic to $\mathbf{Z}\mathrm{/}2\mathbf{Z}\ast \mathcal{O}$ and study certain of its one- and two-dimensional representations. For a periodic continued fraction with period $k$, we give a refined description of the limits of the $k$ different $k$-decimations of its sequence of convergents. We show that for a periodic continued fraction associated to a matrix with eigenvalues of different magnitudes, all $k$ of these limits exist in ${\mathbb{P}}^1(\mathbf{C})$ and a strict majority of them are equal.