Center-freeness of finite-step solvable groups arising from anabelian geometry
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916041956786176 |
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| author | Yamaguchi, Naganori |
| author_facet | Yamaguchi, Naganori |
| contents | Anabelian geometry suggests that, for suitably geometric objects, their étale fundamental groups determine the geometric objects up to isomorphism. From a group-theoretic viewpoint, this philosophy requires rigidity properties, which often follow from their center-freeness of the associated étale fundamental groups. In fact, some profinite groups arising from anabelian geometry are center-free. For any integer $m\geq 2$, we investigate how such center-freeness behaves under passage to the maximal $m$-step solvable quotients. In particular, we show that the maximal $m$-step solvable quotients of the étale and tame fundamental groups of a hyperbolic curve over a separably closed field are torsion-free and center-free. Furthermore, we show that this implies the rigidity property of the $m$-step solvable Grothendieck conjecture. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_07112 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Center-freeness of finite-step solvable groups arising from anabelian geometry Yamaguchi, Naganori Group Theory Algebraic Geometry Primary 14H30, Secondary 14F35, 20E18 Anabelian geometry suggests that, for suitably geometric objects, their étale fundamental groups determine the geometric objects up to isomorphism. From a group-theoretic viewpoint, this philosophy requires rigidity properties, which often follow from their center-freeness of the associated étale fundamental groups. In fact, some profinite groups arising from anabelian geometry are center-free. For any integer $m\geq 2$, we investigate how such center-freeness behaves under passage to the maximal $m$-step solvable quotients. In particular, we show that the maximal $m$-step solvable quotients of the étale and tame fundamental groups of a hyperbolic curve over a separably closed field are torsion-free and center-free. Furthermore, we show that this implies the rigidity property of the $m$-step solvable Grothendieck conjecture. |
| title | Center-freeness of finite-step solvable groups arising from anabelian geometry |
| topic | Group Theory Algebraic Geometry Primary 14H30, Secondary 14F35, 20E18 |
| url | https://arxiv.org/abs/2601.07112 |