SL(2, \Bbb Z), les tresses à trois brins, le tore modulaire et Aut⁺(F₂)

2025/06/24 by Alexis Marin, Marin, Alexis
Mathematics · #01A99 #20E36 #20F36 #20H05 #57K20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2506.19371

openalex publication_date 2025/06/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The action of SL(2, \bf Z) on the integer torus and its quotient by central symmetry and Artin's presentation of three strings braid group B₃, produces a presentation with parabolic generators \pmatrix1& -1\cr 0& 1\cr and \pmatrix1& 0\cr 1& 1\cr. This braided presentation describes the action of the derived group on Poincaré's half plane and its quotient the modular torus, just as Nielsen's theorem giving the group of direct automorphisms of the free group on two generators as semi-direct product, amalgamated on the index 2 subgroup of the center of B₃, of inner automorphisms with B₃.

Citations

• A norm on homology of surfaces and counting simple geodesics

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