Geometry and arithmetic of semi-arithmetic Fuchsian groups
Mikhail Belolipetsky, Gregory Cosac, Cayo Dória, Gisele Teixeira Paula
TL;DR
The paper studies semi-arithmetic Fuchsian groups, a broad class that includes arithmetic groups and many non-arithmetic triangle groups, by introducing a new geometric invariant called stretch $\delta(\Gamma)$ defined via a $\rho_0$-equivariant Lipschitz map built from the Riemannian center of mass. The main result proves finiteness: for fixed $L$, $\mu$, and $r$, there are only finitely many conjugacy classes of semi-arithmetic Fuchsian groups with arithmetic dimension at most $r$, stretch at most $L$, and coarea at most $\mu$, with the arithmetic Margulis lemma providing the key tool. It is shown that $\delta(\Gamma)=1$ for arithmetic groups and for groups admitting modular embeddings, while the construction of infinite families with the same quaternion algebra demonstrates the sharpness of the finiteness results. By bridging spectral, geometric, and arithmetic data, the work offers a quantitative invariant that distinguishes semi-arithmetic groups and has implications for Teichmüller theory and related moduli spaces.
Abstract
Semi-arithmetic Fuchsian groups is a wide class of discrete groups of isometries of the hyperbolic plane which includes arithmetic Fuchsian groups, hyperbolic triangle groups, groups admitting a modular embedding, and others. We introduce a new geometric invariant of a semi-arithmetic group called stretch. Its definition is based on the notion of the Riemannian center of mass developed by Karcher and collaborators. We show that there exist only finitely many conjugacy classes of semi-arithmetic groups with bounded arithmetic dimension, stretch and coarea. The proof of this result uses the arithmetic Margulis lemma. We also show that when stretch is not bounded there exist infinite sequences of such groups.