Area of a word representing the identity

Statement

Let w ∈ F(A) be such that w = 1 in G. Then the area of w, denoted Area(w), is defined as the minimum of the areas of all Van Kampen diagrams with boundary labels w (Van Kampen's lemma says that at least one such diagram exists).

One can show that the area of w can be equivalently defined as the smallest n≥0 such that there exists a representation (♠) expressing w as a product in F(A) of n conjugates of the defining relators.