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Effectiveness in Stallings' Proof of Grushko's Theorem

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Effectiveness in Stallings' Proof of Grushko's Theorem Effectiveness in Stallings' Proof of Grushko's Theorem Paul Synhavsky December 22, 2008 1 Grushko's Theorem The study of free groups and free products plays an essential role in many aspects of both algebra and topology. One of the most useful theorems on this topic is the following due to I. Grushko [Gru40]: Theorem 1.1 (Grushko). Let ' : F −! G1 ∗ G2 be an surjective homomorphism of a free group F onto a free product of groups G1 and G2. Then, there exists a decomposition of F as a free product, F = F1 ∗ F2 such that '(F1) ⊆ G1 and '(F2) ⊆ G2. Algebraically this is useful as it implies every finitely generated group admits a unique decomposition as a free product of indecomposable groups. As an immediate consequence we have additivity of the rank of a free product. Topologically, the connected sum operation, which can be viewed in terms of free products of fundamental groups, plays a critical role in classification of 3-manifolds. In 1940, Grushko proved Theorem 1.1 in an entirely algebraic context. However twenty years later, early in his career, J. Stallings published several topological proofs and extensions of Grushko's theorem. In this paper we examine Stallings' most effective proof of the theorem in its original form [Sta65], as well as some of the theorem's consequences. Here we will primarily restrict ourselves to the case where F is of finite rank and the index set of the free product is finite. By induction, it suffices to consider the case of only two factors. This will allow for an effective, algorithmic decomposition of F . It should be noted that both Grushko's theorem and Stallings' proof extend to the general case of arbitrary rank, although there are details which we do not present. See [Sta65] for an outline of this extension. Instead, we focus on the effective nature and algorithmic aspects of Stallings' method. 1 2 Preliminaries In this section we state some standard definitions and theorems which will be used throughout the paper. Definition 2.1. If X is a subset of a group F , then F is a free group with basis X if, for every group G and every function f : X −! G, there is a unique homomorphism f~ : F −! G extending f such that the diagram below commutes. > F ~ ~ ~ ~ f ~ f X / G Definition 2.2. Let X be a nonempty set and let X−1 be a disjoint set in one-to-one corre- spondence with X via x 7−! x−1. Define the alphabet on X to be X [ X−1. A word on X of −1 length n > 1 is a function w : f1 : : : ; ng ! X [ X . Write a word of length n as follows: ei if w(i) = xi , then e1 en w = x1 ··· xn where xi 2 X and ei = ±1. The empty word, denoted by 1, is a symbol with length defined to be 0. e1 en Definition 2.3. A subword of a word w = x1 ··· xn is either the empty word or a word er es e1 en of the form u = xr ··· xs , where 1 6 r 6 s 6 n. The inverse of a word w = x1 ··· xn is −1 −en −e1 w = xn ··· x1 The words of most interest are reduced words. Definition 2.4. A word w is reduced if w = 1 or if w has no subwords of the form xx−1 or x−1x, where x 2 X. Proposition 2.5 ([Rot02]). Every word w on a set X is equivalent to a unique reduced word. Definition 2.6. Let fGi : i 2 Jg be a family of groups. A free product of the Gi is a group H such that (i) For each index i there is an injective homomorphism ji : Gi −! H. (ii) For every group G and every family of homomorphisms ffi : Gi −! Gg, there is a unique homomorphism : H −! G that extends every fi. Example 2.7. A free group F is a free product of infinite cyclic groups. If F is free on a set X, then hxi is infinite cyclic for each x 2 X. A family of homomor- ~ ~ phisms, fx : hxi −! G determines a function f : X −! G, in particular f(x) = fx(x), which extends to a unique homomorphism : F −! G with jhxi= fx. We have the following normal form statement for free products. 2 Theorem 2.8 (Normal Form, [Rot02]). If w 2 ∗Gi and w 6= 1, then w has a unique factorization w = g1g2 ··· gn where each gi 6= 1 and adjacent factors lie in distinct Gi. If we regard the elements of ∗Gi as reduced words, the theorem follows. The spaces we consider will always be CW-complexes. Given a presentation of any group G, one can build a CW-complex X so that π1(X) ' G. Recall a CW-complex is a structure on a Hausdorff space X given by an ascending sequence of closed subspaces X0 ⊂ X1 ⊂ X2 ··· which satisfy the following conditions: 1. X0 has the discrete topology. n n−1 2. For n > 1, X is obtained from X by adjoining a collection of n-cells via attaching maps. i 3. X is the union of the subspaces X for i > 0. 4. The space X and the subspaces Xi all have the weak topology. The proof of the main theorem depends on an application of the Seifert-van Kampen theorem [Mas67]: Theorem 2.9 (Seifert-van Kampen). Suppose a CW complex X can be expressed as the union of two non-empty, path connected subcomplexes A and B, where A \ B is simply connected. Then π1(X) ' π1(A) ∗ π1(B): 3 Example To gain a feeling of the flavor of the proof of the theorem, we give an example which demon- strates the effectiveness of Stallings' method. In this example, we consider the free product of two groups G1 ∗ G2 where G1 is a cyclic 2 group of order two with presentation ha : a i and G2 is cyclic of order three with presention hb : b3i. Let F be a free group on two generators x and y. Consider the map ' : F (x; y) −! ha : a2i ∗ hb : b3i '(x) 7−! ab2 '(y) 7−! aba: For each i = 1; 2, construct a two-dimensinal CW complex Bi with a single vertex vi that is determined by the presentation of Gi. We then have π1(Bi; vi) = Gi. Let Y be the quotient space of B1 [ B2 obtained by identifying the vi to a single vertex v. Then π1(Y; v) = G1 ∗ G2: 3 3.1 Construction of X Construct a space X composed of two 1-spheres, one for each generator of F , identified at a base point, as follows. Let g 2 fx; yg be a generator of F , and consider the representation of '(g) in G1 ∗ G2. This can be expressed as a reduced word in fa; bg. Let Sg denote the 1-sphere corresponding to g. Divide Sg into n segments by n vertices, one for each letter in '(g). Now, define a map f : Sg −! Y so that the restriction of f to each segment is a path in some Bi representing the generator of Gi. In this fashion, f defines a labeling scheme for X so that reading around Sg from the base point is exactly '(g) 2 G1 ∗ G2. It is helpful to think of each segment in X as having a color corresponding to the Bi into which it is mapped. Say for example, the segments mapped into G1 are red and those mapped into G2 are blue. We color all the vertices both red and blue, simultaneously. X: Y: a2 b1 v1 v2 a a1 b2 f v0 a a 3 2 b b3 v4 v3 b3 Figure 1: The spaces X and Y . Now we have that π1(X) = F (x; y) and the union of the maps f defines a unique map f : X ! Y such that the induced homomorphism f∗ : π1(X) −! π1(Y ) is equivalent to our original surjection '. Let Ai denote the union of all the vertices of X along with the segments which are mapped into Bi. Then f(Ai) ⊂ Bi and the intersection of the Ai consists of all the vertices of X. 4 A 1 A 1 A 2 A 2 b1 v1 v2 v1 v2 v1 v2 a1 b2 v0 v0 v0 a3 a2 v4 v3 v4 v3 v4 v3 b3 Figure 2: The subcomplexes A1 and A2 and their intersection. In addition to the standard results from the previous section, in the following exposition the notion of a binding tie will be of crucial importance. Definition 3.1. Let X and Y be a pair of connected CW-complexes that are each the union of two subcomplexes, X = A1 [ A2 and Y = B1 [ B2. Let f : X ! Y be a map that sends the n-skeleton of X to the n-skeleton of Y so that f(Ai) ⊆ Bi. A path p in X will be called a binding tie if is entirely contained in one Ai while its end points lie in different components of the intersection A1 \ A2, and fp is homotopic to a path in B1 \ B2 in the Bi to which it was mapped. One of the main results in this paper, which allowed Stallings to prove Grushko's theorem in this topological fashion is the following lemma, which we will prove in Section 5. Lemma 3.2. Let X and Y be a pair of connected CW-complexes that are each the union of two subcomplexes, X = A1 [ A2 and Y = B1 [ B2.
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