The groups ${\Gamma }_n^4$, braids, and $3$-manifolds

Abstract

We introduce a family of groups ${\Gamma }_n^k$ 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 ${\Gamma }_n^4$ we mention the Ptolemy relation, Pentagon relation, cluster algebra, Stasheff polytope.