Classification of the conjugacy classes of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$
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| Format: | Preprint |
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2023
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| _version_ | 1866915066330218496 |
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| author | Táfula, Christian |
| author_facet | Táfula, Christian |
| contents | 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 $α\in \widetilde{\mathrm{SL}}_2(\mathbb{R})$, 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 $\overlineα$ 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]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_08617 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classification of the conjugacy classes of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$ Táfula, Christian Group Theory 20E45, 22E46 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 $α\in \widetilde{\mathrm{SL}}_2(\mathbb{R})$, 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 $\overlineα$ 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]. |
| title | Classification of the conjugacy classes of $\widetilde{\mathrm{SL}}(2,\mathbb{R})$ |
| topic | Group Theory 20E45, 22E46 |
| url | https://arxiv.org/abs/2304.08617 |