6.2 Decomposing groups which act on trees (Bass-Serre Theory) 157 ThusBS(m, n) is the fundamental group of a circle of groups (G, Γ) whereΓ has one vertex and one edge, and both the vertex group and the edge group are infinite cyclic. The homomorphisms φk : Z → Z, 1 → k, are induced by covering projections S1 → S1 when k ≥1. Since all these covering pro- jections lift to homeomorphisms of R, there is an obvious homeomorphism h: U →Y ×R where Y is the Bass-Serre tree of (G, Γ) andU is as in the above construction. Indeed, usinghto makeY ×Rinto aGspace, we have a commutative diagram ofG-spaces:
U h //
s@@@@@
@@ Y ×R
projection
{{xxxxxxxxx
Y
This is a topological description of U. But U is also a CW complex, the universal cover of Tot(X, Γ), which in this case is the presentation complex of BS(m, n). Exercise 8 of Sect. 3.2 asked for a description of the cells of the CW complexU in the case of BS(1,2). It is instructive to “see” the topological productY ×Rin this CW complex.
The construction ofY from (G, Γ) is inverse to 6.2.7 in the following sense.
In 6.2.7 we started withGacting onY, together with a maximal treeT ⊂Γ, and produced a decomposition of G as π1( ¯G, Γ;T) whose vertex and edge groups are stabilizers of certain vertices and edges (determined by choices of T˜, of{e˜|e∈E0∪E1}and of orientation ofΓ). Conversely, we have sketched how one starts with (G, Γ) andT, and produces an action ofG:=π1(G, Γ;T) on a treeY, such that (with appropriate choices) the corresponding graph of groups ( ¯G, Γ) is isomorphic to (G, Γ). Indeed, it can be shown that: starting with the G-tree Y, obtaining ( ¯G, Γ), and then constructing a π1( ¯G, Γ;T)- tree ¯Y as above, there is an equivariant isomorphism from ¯Y to Y (i.e., an isomorphism commuting with the actions of the groups). For this reason, the theories of graphs of groups and of groups acting rigidly on trees are considered to be equivalent. For more details the reader is referred to [140].
A more algebraic version is described in [141] and [142].
of groups are useful in computing fundamental groups via a generalization of the Seifert-Van Kampen Theorem 3.1.18, as we now explain.
Let {Xα}α∈A be a cover of the path connected CW complex X by path connected subcomplexes such that no point ofX lies in more than two of the Xα’s. LetΓ be the graph whose vertex set isA, having an edgeeαβγ joining α=β for each path componentYαβγ ofXα∩Xβ. ThenΓ is a path connected graph. OrientΓ.
By an obvious extension of 3.1.13, choose a maximal tree ¯T in X such that each ¯T∩Xα and each ¯T∩Yαβγ is a maximal tree in that subcomplex.
Let φαβγ : π1(Yαβγ,T¯∩Yαβγ) → π1(Xβ,T¯∩Xβ) be induced by inclusion.
Then we have a generalized graph of groups (G, Γ) whose vertex group at α is π1(Xα,T¯∩Xα), whose edge group over the edge labeled by Yαβγ is π1(Yαβγ,T¯∩Yαβγ), and whose structural homomorphisms for that edge are φαβγ andφβαγ. Pick a maximal treeT inΓ.
Theorem 6.2.11.[Generalized Van Kampen Theorem] Under these hypothe- ses the fundamental group ofX is isomorphic toπ1(G, Γ;T).
Proof. Let (X, Γ) be the generalized graph of path connected CW complexes havingXαoverα,Yαβγover the edge joiningαtoβso indexed, and inclusions as structural maps. Let Tot(X, Γ) be the total complex. The proof of 6.2.2 extends to show that π1(G, Γ;T) is isomorphic to the fundamental group of Tot(X, Γ). The space Tot(X, Γ) presents itself naturally as a subcomplex of X×Γ, consisting ofXαover the vertexαandYαβγ×eαβγ over each edgeeαβγ
joiningαtoβ. Projection on theXfactor gives us a mapp: Tot(X, Γ) → X.
Consider the commutative diagram Tot(X, Γ)
id oo ? _∪(Yαβγ×eαβγ) id //
id
∪(Yαβγ×eαβγ)
p|
Tot(X, Γ)oo ? _∪(Yαβγ×eαβγ)
p| //∪Yαβγ
The hypothesis that no point ofX lies in more than two of theXα’s implies that the spacesYαβγ are pairwise disjoint and hence the indicated map p| is a homotopy equivalence. The map of adjunction spaces coming from 4.1.7 is preciselyp, which is therefore a homotopy equivalence.
Remark 6.2.12.As with ordinary graphs of groups there is an obvious epi- morphism π1(G, Γ;T) π1(Γ, T). Since π1(Γ, T) is free this gives a lower bound for the number of generators ofπ1(X, v) in 6.2.11. In fact, since every epimorphism onto a free group has a right inverse we see that π1(Γ, T) is a retract11 ofπ1(G, Γ;T).
11The definition ofretract in any category is analogous to that given in Sect. 1.3 for Spaces.
6.2 Decomposing groups which act on trees (Bass-Serre Theory) 159 Remark 6.2.13.We end by discussing the relationship between graphs of groups and generalized graphs of groups. Let (G, Γ) be a generalized graph of groups, letT be a maximal tree inΓ and letG:=π1(G, Γ;T). Proceeding as in the “inverse construction” part of this section, one forms a graph of pointed CW complexes (X, Γ), identifying Gwithπ1(Tot(X, Γ), x(v)); the universal cover of Tot(X, Γ) is denoted by U. The space U is built out of covering spaces, no longer universal covers, of vertex complexes and edge complexes, and, as before, one obtains aG-treeY from this situation. The quotientG\Y is a copy of Γ. The vertex and edge stabilizers of this new graph of groups ( ¯G, Γ) are the images inGof the vertex and edge groups ofG. This new de- composition ofG may be quite uninteresting. For example, takeΓ to be an edge, take both vertex groups to be trivial and take the edge group to beZ (as happens when the Seifert-Van Kampen Theorem is applied to S2, using the two hemispheres as the “vertex spaces”). Then all the groups in ¯G are trivial.
Source Notes:The theory described in this section appeared in [141], translated into English as [142]. A more topological presentation, on which this section is based, appeared in [140].
Exercises
1. Prove 6.2.5.
2. Referring to Case 1 at the beginning of this section, describe the Bass-Serre tree in the case of a free product with amalgamation in which the monomorphism φ+e is an isomorphism, and in the case whereφ+e andφ−e are both isomorphisms.
3. Describe the Bass-Serre tree of the circle of groups corresponding to the given presentation of the Baumslag-Solitar groupBS(m, n).
4. In an obvious way,Zcan be decomposed as the fundamental group of a graph of groups, with one vertex and one edge, the vertex and edge groups being trivial.
Thus ifα:G→Zis an epimorphism, there is a corresponding decomposition of Gwith vertex and edge groups isomorphic to ker(α). Give an example withG finitely presented such that ker(α) is not finitely generated; give another graph of groups decomposition of the same groupGwith one vertex and one edge so that the vertex and edge groups are finitely generated.
5. LetG=A∗
C(B∗
ED) be a decomposition ofGusing free products with amalga- mation. Under what conditions does this decompose as the fundamental group of a graph of groups with graph•–•–•and with vertex groups isomorphic to A,B andDand edge groups isomorphic toCandE?
6. Prove 6.2.2 whenX(w) orX(e) has more than one vertex.
7. Let ¯G=G∗φ be an HNN extension whereφ is an automorphism ofG. Prove that ¯Gis a semidirect product ofGandZ.