A remark on the existence of equivariant functions

Abstract

Let $\Gamma$ be a Fuchsian group in ${\rm SL}_2(\mathbb{R})$. In this note, we discuss the existence of $\rho$-equivariant functions for a two-dimensional representation $\rho$ of $\Gamma$. This assertion was first stated by Saber and Sebbar in 2020, and this note partially fills a gap of their statement by proving the assertion for a certain class of Fuchsian groups such as conjugates of subgroups of ${\rm SL}_2(\mathbb{Z})$.