Reversibility in the Seifert-fibered spaces

An element a in a group Γ is called reversible if there exists g ∈ Γ 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. In the last section we apply this classification to the braid groups, particularly to the braid group on $3$ strands.