  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.

Group Theory;Geometric Topology;Metric Geometry;Number Theory