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On Ubiquitous groups \Gamma_n^k



2023 05 25

17:00 (GMT+0)


Vassily Olegovich Manturov, Moscow Institute of Physics and Technology


We shall introduce a family of groups $\Gamma_n^k$ for integer parameters n>k.

These groups initially arose in relation to braids on 2-surfaces (G_{n}^{4} and their analogues for higher-dimensional surfaces, k>4, I.M.Nikonov).

The crucial relation for these group is the well-known pentagon identity.
This identity has been widely studied by various authors in different contexts
(say, in the book by Gelfand-Kapranov-Zelevinsky in relation to triangulations),
and it immediately appears that:
Invariants of braids give rise to invariants of 3-manifolds and invariants of
3-manifolds give rise to invariants of (surface) braids.

The book of Gelfand-Kapranov-Zelevinsky dealt with (spaces of) triangulations from
topological, combinatorial, functorial and other points of view, but the authors overlooked
groups which appeared naturally in this context.

Cluster algebras, Coxeter groups, Associahedra (Stasheff polytopes) all play crucial
roles in science because they are all related to (and connect) various branches of
The groups $\Gamma_{n}^{k}$ are related to at least these three notions.

Apart from low-dimensional topology, we list several branches of mathematics
where the groups $\Gamma$ naturally appear:

triangulations (ideal triangulations),
braids, knots, manifolds, and their invariants,
cluster algebras,
tropical geometry,
s-t duality,
Drinfeld's associators,
plane arrangements,
and, probably, many other areas still to be discovered and to dream about
like dessins d'enfant,
hyperbolic structures.

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