Modular Embeddings and Rigidity for Fuchsian Groups

TL;DR

This paper establishes a rigidity theorem for semi-arithmetic Fuchsian groups with modular embeddings, showing that isomorphisms respecting congruence subgroups are induced by inner automorphisms of PGL(2,R).

Contribution

It proves a new rigidity result for semi-arithmetic Fuchsian groups with modular embeddings, linking group isomorphisms to inner automorphisms.

Findings

Group isomorphisms respecting congruence subgroups are inner automorphisms.

Rigidity holds for semi-arithmetic Fuchsian groups with modular embeddings.

The result connects algebraic properties with geometric automorphisms.

Abstract

We prove a rigidity theorem for semi-arithmetic Fuchsian groups: If $\Gamma_1$, $\Gamma_2$ are two semi-arithmetic lattices in $\mathrm{PSL}(2,\mathbb{R})$ virtually admitting modular embeddings and $f\colon\Gamma_1\to\Gamma_2$ is a group isomorphism that respects the notion of congruence subgroups, then $f$ is induced by an inner automorphism of $\mathrm{PGL}(2,\mathbb{R})$.