In this note, we classify the conjugacy classes of $\widetilde{ \mathrm{ SL}}_2(\mathbb{ R})$, the universal covering group of $\mathrm{ PSL}_2(\mathbb{ R})$. For any non-central element $\alpha \in \widetilde{ \mathrm{ SL}}_2(\mathbb{ R})$, we show that its conjugacy class may be determined by three invariants: (i) Trace: the trace (valued in the set of positive real numbers $\mathbb{ R}_{ +}$) of its image $\overline{ \alpha}$ in $\mathrm{ PSL}_2(\mathbb{ R})$; (ii) Direction type: the sign behavior of the induced self-homeomorphism of $\mathbb{ R}$ determined by the lifting $\widetilde{ \mathrm{ SL}}_2(\mathbb{ R}) \curvearrowright \mathbb{ R}$ of the action $\mathrm{ PSL}_2(\mathbb{ R}) \curvearrowright \mathbb{ S}^{ 1}$; (iii) The function $\ell^{ \sharp}$: a conjugacy invariant length function introduced by S. Mochizuki [Res. Math. Sci. 3 (2016), 3:6].
Categories: math.GR
