Gnk-groups for simplicial complexes and the word problem on G2(K)

We introduce a natural [Formula: see text]-type construction of groups on any given simplicial complex [Formula: see text] called [Formula: see text] as a generalization of the notion of groups [Formula: see text], which gives a functor from the category of simplicial complexes and injective simplicial maps to the category of groups. We prove that the word problem on [Formula: see text] is solvable.