Reversible and other generalised torsion elements in Seifert-fibered spaces

An element $a$ in a group $\Gamma$ is called \emph{reversible} if there exists $g \in \Gamma$ such that $gag^{-1}=a^{-1}$. The reversible elements are also known as `real elements' or `reciprocal elements' in literature. In this paper, we classify the reversible elements in Fuchsian groups, and use this classification to find all reversible elements in a Seifert-fibered group. We then apply the classification to the braid groups, particularly to the braid group on $3$ strands. We further study generalised 3-torsion elements in PSL(2,Z), and use this to analyse the existence of generalised 3-torsion elements in Seifert-fibered spaces in general, and braid groups on 3 strands in particular.