  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.
