Grushko decomposition theorem

Statement

A useful consequence of the original Grushko theorem is the so-called Grushko decomposition theorem. It asserts that any nontrivial finitely generated group G can be decomposed as a free product

G = A₁∗A₂∗...∗A_r∗F_s, where s ≥ 0, r ≥ 0,

where each of the groups A_i is nontrivial, freely indecomposable (that is, it cannot be decomposed as a free product) and not infinite cyclic, and where F_s is a free group of rank s; moreover, for a given G, the groups A₁, ..., A_r are unique up to a permutation of their conjugacy classes in G (and, in particular, the sequence of isomorphism types of these groups is unique up to a permutation) and the numbers s and r are unique as well.

More precisely, if G = B₁∗...∗B_k∗F_t is another such decomposition then k = r, s = t, and there exists a permutation σ∈S_r such that for each i=1,...,r the subgroups A_i and B_σ(i) are conjugate in G.

The existence of the above decomposition, called the Grushko decomposition of G, is an immediate corollary of the original Grushko theorem, while the uniqueness statement requires additional arguments (see, for example).

Algorithmically computing the Grushko decomposition for specific classes of groups is a difficult problem which primarily requires being able to determine if a given group is freely decomposable. Positive results are available for some classes of groups such as torsion-free word-hyperbolic groups, certain classes of relatively hyperbolic groups, fundamental groups of finite graphs of finitely generated free groups and others.

Grushko decomposition theorem is a group-theoretic analog of the Kneser prime decomposition theorem for 3-manifolds which says that a closed 3-manifold can be uniquely decomposed as a connected sum of irreducible 3-manifolds.