Hecke Triangle Groups and Special Hyperbolic Elements

We study the action of the Hecke triangle groups $G_q$ on ${\lambda }_q\mathbb{Q}({\lambda }_q^2)\cup \{\mathrm{\infty }\}$ with ${\lambda }_q=2\cos (\pi \mathrm{/}q)$. When $q=18$, we show the existence of infinitely many distinct orbits of fixed points of special hyperbolic elements of $G_q$. We also find new orbits for several other values of $q$. These results provide new examples of special affine pseudo-Anosov homeomorphisms on the unfoldings of regular $q$-gons. In particular, on the unfolding of the regular $18$-gon, there are infinitely many distinct Veech group orbits of directions invariant under a special affine pseudo-Anosov.