A remark on the existence of equivariant functions

TL;DR

This paper investigates the existence of $ ho$-equivariant functions for certain Fuchsian groups, extending previous results by proving the assertion for conjugates of subgroups of ${ m SL}_2( obreak{ ext{f Z}})$, thus filling a gap in earlier work.

Contribution

It provides a partial proof of the existence of $ ho$-equivariant functions for specific classes of Fuchsian groups, notably conjugates of subgroups of ${ m SL}_2( obreak{ ext{f Z}})$, addressing a previously unproven case.

Findings

Proves existence of $ ho$-equivariant functions for conjugates of subgroups of ${ m SL}_2( obreak{ ext{f Z}})$

Fills a gap in the previous assertion by Saber and Sebbar (2020)

Extends understanding of equivariant functions for certain Fuchsian groups

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 ${\rm SL}_2(\mathbb{Z})$.