Classification of the conjugacy classes of SL˜(2,R)

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