Hecke Triangle Groups and Special Hyperbolic Elements

Abstract

We study the action of the Hecke triangle groups $G_q$ on $\hat{I}{»}_q\mathbb{Q}(\hat{I}{»}_q^2)\cup \{\infty \}$ with $\hat{I}{»}_q=2\cos (\ddot{I}€/q)$. 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.

16 pages