Classification of the conjugacy classes of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$

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].