The groups Γ_n⁴, braids, and 3-manifolds

2023/05/10 by Vassily Olegovich Manturov, Manturov, Vassily Olegovich, Igor Nikonov +1
Mathematics · #13F60 #20F36 #57K20 #57K31 #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2305.06316

openalex publication_date 2023/05/10 · openalex created_date 2023/05/12 · openalex updated_date 2026/07/28

Abstract

We introduce a family of groups Γ_n^k for integer parameters n>k. These groups originate from discussion of braid groups on 2-surfaces. On the other hand, they turn out to be related to 3-manifolds (in particular, they lead to new relationships between braids and manifolds), triangulations (ideal triangulations) cluster algebras, dynamics of moving points, quivers, hyperbolic structures, tropical geometry, and, probably, many other areas still to be discovered. Among crucial reason of this importance of groups Γ_n⁴ we mention the Ptolemy relation, Pentagon relation, cluster algebra, Stasheff polytope.

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