Given a subring $A$ of $\mathbb{R}$, we can consider the set $\mathrm{PSL}_{2}(A)$ of elements in $\mathrm{PSL}_{2}(\mathbb{R})$ with entries in $A$ and the determinant of associated matrix is equal to one.

We define a hyperbolic elements from $\mathrm{PSL}_{2}(A)$ as elements which the trace of the associated matrix is greater than two, in absolute value.

Let us denote the set of all fixed points of all hyperbolic elements from $\mathrm{PSL}_{2}(A)$ by $\mathcal{P}_{A}$. Notice that $\mathcal{P}_{A}\subseteq \mathbb{R}\cup\{\infty\}$. If $A$ is a proper subring of $\mathbb{R}$, does it exist some result about $\mathcal{P}_{A}$ being a group?