Basic PU(1,1)-representations of the hyperelliptic group are discrete

TL;DR

This paper proves that basic PU(1,1)-representations of the hyperelliptic group are exactly those that are discrete and faithful, confirming part of a conjecture related to the Poincaré disc.

Contribution

It establishes a precise criterion linking basic representations to discreteness and faithfulness for the hyperelliptic group, advancing understanding in geometric group theory.

Findings

Basic PU(1,1)-representations are discrete and faithful

Partial proof of a conjecture by Anan'in and Gonçalves

Clarifies the structure of hyperelliptic group representations

Abstract

We show that a PU(1,1)-representation of the hyperelliptic group $H_{n}$ is basic if and only if it is discrete and faithful, thus partially proving a conjecture by S. Anan'in and E. Bento Gon\c{c}alves in the case of the Poincar\'{e} disc.