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In [ 1 ], Barot and Marsh presented an explicit construction of presentation of a finite Weyl group   $W$   by any initial seed of corresponding cluster algebra, that is, by any diagram mutation-equivalent to an orientation of a Dynkin diagram with Weyl group   $W$  . We obtain similar presentations for all affine Coxeter groups. Furthermore, we generalize the construction to the settings of diagrams arising from unpunctured triangulated surfaces and orbifolds, which leads to presentations of corresponding groups as quotients of numerous distinct Coxeter groups.

Coxeter Groups and their Quotients Arising from Cluster Algebras