Reversible and other generalised torsion elements in Seifert-fibered spaces

An element $a$ in a group $\mathrm{\Gamma }$ is called \emph{reversible} if there exists $g\in \mathrm{\Gamma }$ such that $ga{g}^{-1}=a^{-1}$. 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.