The normal forms theorem

Statement

Let g be an element of G = π₁(A, T) represented as a product of the form

$$g=a_0e_1a_1\dots e_na_n,$$

where e₁, ..., e_n is a closed edge-path in A with the vertex sequence v₀, v₁, ..., v_n = v₀ (that is v₀=o(e₁), v_n = t(e_n) and v_i = t(e_i) = o(e_i+1) for 0 < i < n) and where $a_i\in A_{v_i}$ for i = 0, ..., n.

Suppose that g = 1 in G. Then

• either n = 0 and a₀ = 1 in $A_{v_0}$,
• or n > 0 and there is some 0 < i < n such that e_i+1 = $\overline{e_{i}}$ and $a_i\in \omega_{e_i}(A_{e_i})$.

The normal forms theorem immediately implies that the canonical homomorphisms A_v → π₁(A, T) are injective, so that we can think of the vertex groups A_v as subgroups of G.

Higgins has given a nice version of the normal form using the fundamental groupoid of a graph of groups. This avoids choosing a base point or tree, and has been exploited by Moore.