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Keywords:
reversibility; reversible element; reversing symmetry; quaternionic Möbius transformation; algebraic characterization; conjugacy invariant
reversibility; reversible element; reversing symmetry; quaternionic Möbius transformation; algebraic characterization; conjugacy invariant
Summary:
An element of a group is called reversible if it is conjugate to its inverse. While reversibility in the quaternionic Möbius group ${\rm PSL}(2,\mathbb {H})$ has traditionally been studied using geometric and dynamical methods, we develop a purely algebraic approach. We obtain an explicit, computable criterion for the reversibility of a quaternionic Möbius transformation, expressed solely in terms of the entries of a matrix representative. More precisely, we prove that \[ [A]\in {\rm PSL}(2,\mathbb {H}) \ \text {is reversible} \Longleftrightarrow \beta _A^{2}=\delta _A^{2}, \] where $\beta _A$ and $\delta _A$ are real conjugacy invariants associated with a lift $A\in {\rm SL}(2,\mathbb {H})$. Furthermore, we give a complete characterization of reversing symmetries of reversible elements in ${\rm SL}(2,\mathbb {H})$ and ${\rm PSL}(2,\mathbb {H})$.
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