Hecke Triangle Groups and Special Hyperbolic Elements

Karl Winsor

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arXiv · 2605.30064

Hecke Triangle Groups and Special Hyperbolic Elements

Karl Winsor

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Abstract

We study the action of the Hecke triangle groups $G_q$ on $\lambda_q \mathbb{Q}(\lambda_q^2) \cup \{\infty\}$ with $\lambda_q = 2 \cos (\pi / q)$. 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.

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BibTeX RIS

Karl Winsor. 2026-05-28. Hecke Triangle Groups and Special Hyperbolic Elements. https://arxiv.org/abs/2605.30064

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