Grushko theorem

In the mathematical subject of group theory, the Grushko theorem or the Grushko-Neumann theorem is a theorem stating that the rank (that is, the smallest cardinality of a generating set) of a free product of two groups is equal to the sum of the ranks of the two free factors. The theorem was first obtained in a 1940 article of Grushko^[1] and then, independently, in a 1943 article of Neumann.^[2]

Contents 1 Statement of the theorem 2 History and generalizations 3 Grushko decomposition theorem 4 Sketch of the proof using Bass-Serre theory 5 See also 6 Notes

Statement of the theorem

Let A and B be finitely generated groups and let A∗B be the free product of A and B. Then

rank(A∗B) = rank(A) + rank(B).

It is obvious that rank(A∗B) ≤ rank(A) + rank(B) since if X is a finite generating set of A and Y is a finite generating set of B then X∪Y is a generating set for A∗B and that |X∪Y|≤|X| + |Y|. The opposite inequality, rank(A∗B) ≥ rank(A) + rank(B), requires proof.

There is a more precise version of Grushko's theorem in terms of Nielsen equivalence. It states that if M = (g₁, g₂, ..., g_n) is an n-tuple of elements of G = A∗B such that M generates G, <g₁, g₂, ..., g_n> = G, then M is Nielsen equivalent in G to an n-tuple of the form

M' = (a₁, ..., a_k, b₁, ..., b_n−k) where {a₁, ..., a_k}⊆A is a generating set for A and where {b₁, ..., b_n−k}⊆B is a generating set for B. In particular, rank(A) ≤ k, rank(B) ≤ n − k and rank(A) + rank(B) ≤ k + (n − k) = n. If one takes M to be the minimal generating tuple for G, that is, with n = rank(G), this implies that rank(A) + rank(B) ≤ rank(G). Since the opposite inequality, rank(G) ≤ rank(A) + rank(B), is obvious, it follows that rank(G)=rank(A) + rank(B), as required.

History and generalizations

After the original proofs of Grushko (1940) and Neumann(1943), there were many subsequent alternative proofs, simplifications and generalizations of Grushko's theorem. A close version of Grushko's original proof is given in the 1955 book of Kurosh.^[3]

Like the original proofs, Lyndon's proof (1965)^[4] relied on length-functions considerations but with substantial simplifications. A 1965 paper of Stallings ^[5] gave a greatly simplified topological proof of Grushko's theorem.

A 1970 paper of Zieschang^[6] gave a Nielsen equivalence version of Grushko's theorem (stated above) and provided some generalizations of Grushko's theorem for amalgamated free products. Scott (1974) gave another topological proof of Grushko's theorem, inspired by the methods of 3-manifold topology^[7] Imrich (1984)^[8] gave a version of Grushko's theorem for free products with infinitely many factors.

Modern techniques of Bass-Serre theory, particularly the machinery of foldings for group actions on trees and for graphs of groups provide a relatively straightforward proof of Grushko's theorem (see, for example ^[9][10]).

Grushko's theorem is, in a sense, a starting point in Dunwoody's theory of accessibility for finitely generated and finitely presented groups. Since the ranks of the free factors are smaller than the rank of a free product, Grushko's theorem implies that the process of iterated splitting of a finitely generated group G as a free product must terminate in a finite number of steps (more precisely, in at most rank(G) steps). There is a natural similar question for iterating splittings of finitely generated groups over finite subgroups. Dunwoody proved that such a process must always terminate if a group G is finitely presented^[11] but may go on forever if G is finitely generated but not finitely presented.^[12]

An algebraic proof of a substantial generalization of Grushko's theorem using the machinery of groupoids was given by Higgins (1966).^[13] Higgins' theorem starts with groups G and B with free decompositions G = ∗_i G_i, B = ∗_i B_i and f : G → B a morphism such that f(G_i) = B_i for all i. Let H be a subgroup of G such that f(H) = B. Then H has a decomposition H = ∗_i H_i such that f(H_i) = B_i for all i. Full details of the proof and applications may also be found in .^[14]

Grushko decomposition theorem

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^[15]).

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,^[16] fundamental groups of finite graphs of finitely generated free groups^[17] 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.^[18]

Sketch of the proof using Bass-Serre theory

The following is a sketch of the proof of Grushko's theorem based on the use of foldings techniques for groups acting on trees (see ^[9][10] for complete proofs using this argument).

Let S={g₁,....,g_n} be a finite generating set for G=A∗B of size |S|=n=rank(G). Realize G as the fundamental group of a graph of groups Y which is a single non-loop edge with vertex groups A and B and with the trivial edge group. Let be the Bass-Serre covering tree for Y. Let F=F(x₁,....,x_n) be the free group with free basis x₁,....,x_n and let φ₀:F → G be the homomorphism such that φ₀(x_i)=g_i for i=1,...,n. Realize F as the fundamental group of a graph Z₀ which is the wedge of n circles that correspond to the elements x₁,....,x_n. We also think of Z₀ as a graph of groups with the underlying graph Z₀ and the trivial vertex and edge groups. Then the universal cover of Z₀ and the Bass-Serre covering tree for Z₀ coincide. Consider a φ₀-equivariant map so that it sends vertices to vertices and edges to edge-paths. This map is non-injective and, since both the source and the target of the map are trees, this map "folds" some edge-pairs in the source. The graph of groups Z₀ serves as an initial approximation for Y.

We now start performing a sequence of "folding moves" on Z₀ (and on its Bass-Serre covering tree) to construct a sequence of graphs of groups Z₀, Z₁, Z₂, ...., that form better and better approximations for Y. Each of the graphs of groups Z_j has trivial edge groups and comes with the following additional structure: for each nontrivial vertex group of it there assigned a finite generating set of that vertex group. The complexity c(Z_j) of Z_j is the sum of the sizes of the generating sets of its vertex groups and the rank of the free group π₁(Z_j). For the initial approximation graph we have c(Z₀)=n.

The folding moves that take Z_j to Z_j+1 can be of one of two types:

• folds that identify two edges of the underlying graph with a common initial vertex but distinct end-vertices into a single edge; when such a fold is performed, the generating sets of the vertex groups and the terminal edges are "joined" together into a generating set of the new vertex group; the rank of the fundamental group of the underlying graph does not change under such a move.
• folds that identify two edges, that already had common initial vertices and common terminal vertices, into a single edge; such a move decreases the rank of the fundamental group of the underlying graph by 1 and an element that corresponded to the loop in the graph that is being collapsed is "added" to the generating set of one of the vertex groups.

One sees that the folding moves do not increase complexity but they do decrease the number of edges in Z_j. Therefore the folding process must terminate in a finite number of steps with a graph of groups Z_k that cannot be folded any more. It follows from the basic Bass-Serre theory considerations that Z_k must in fact be equal to the edge of groups Y and that Z_k comes equipped with finite generating sets for the vertex groups A and B. The sum of the sizes of these generating sets is the complexity of Z_k which is therefore less than or equal to c(Z₀)=n. This implies that the sum of the ranks of the vertex groups A and B is at most n, that is rank(A)+rank(B)≤rank(G), as required.

See also

• Bass-Serre theory
• Generating set of a group

Notes