Classification of the conjugacy classes of \(\widetilde{\operatorname{SL}} (2, \mathbb{R})\) (Q6656076)

Let \(\widetilde{\mathrm{SL}}(2,\mathbb{R})\) be the universal covering group of \(\mathrm{PSL}_{2}(\mathbb{R})\). The goal of the paper under review is to classify the conjugacy classes of the non-central elements \(\alpha \in \widetilde{\mathrm{SL}}(2,\mathbb{R})\). In particular the author shows that the conjugacy of \(\alpha\) may be determined by three invariants: (i) the trace of the image of \(\alpha\) 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 \(\mathrm{PSL}_{2}(\mathbb{R}) \curvearrowright \mathbb{S}^{1}\) and (iii) the function \(\ell^{\sharp}\): a conjugacy invariant length function introduced by \textit{S. Mochizuki} in [Res. Math. Sci. 3, Paper No. 6, 21 p. (2016; Zbl 1401.14134)]. The direction type and \(\ell^{\sharp}\) function are invariant under conjugation in \(\widetilde{\textrm{Homeo}}_{+}(\mathbb{S}^{1})\).