Decomposing groups which act on trees (Bass-Serre Theory)Theory)

The Borel Construction and Bass-Serre Theory

6.2 Decomposing groups which act on trees (Bass-Serre Theory)Theory)

Proof. By 6.1.1,q⁻¹(V¹) is a subcomplex ofZ containingZ¹. By hypothesis in (i), for eachw∈E0,π1(Xw, xw) is finitely generated. By 4.1.14, eachXwis homotopy equivalent to a CW complexX_w having finite 1-skeleton. By 3.1.13 and 4.1.9, when e ∈ E1, Xe is homotopy equivalent to a CW complex X_e having one vertex. By applying 6.1.4 to the stack q : Z → V, we obtain a stack q :Z →V in whichZ is homotopy equivalent toZ, and the fiber of q overe[resp. overw] isX_e [resp.X_w].

The CW complexesZand (q)⁻¹(V¹) have the same 1-skeleton. It is finite because: (a)Γ⁰ is finite and ((q)⁻¹(V⁰))¹is the finite graph

w∈E₀

(X_w)¹; (b) V¹ is finite and (sinceX_e×B¹is a product and X_e has one vertex) there is only one 1-cell of (q)⁻¹(V¹) mapped onto each 1-cell ofΓ. (Note that (1-cell ofX_e)×(vertex ofB¹) does not give a new 1-cell of the adjunction complex.) ThusZ has finite 1-skeleton. By 3.1.17,G∼=π1(Z, z)∼=π1(Z, z) is finitely generated. Thus (i) holds.

The proof of (ii) is similar. We replace eachXw byX_w , whose 2-skeleton is the wedge of a finite 2-complex and a bouquet of 2-spheres (see 4.1.14); for e∈E1, we replaceXe by X_e having finite 1-skeleton; fore∈E2 we replace Xe by X_e having a single vertex. The resulting CW complex Z has finite 1-skeleton (for reasons already explained) and possibly infinite 2-skeleton.

Indeed, its 2-cells are of three kinds: 2-cells ofX_w wherew∈E₀, cells coming from (1-cells ofX_e)×B¹fore∈E₁, and cells coming from (vertex ofX_e)×B² fore∈E₂. Thus all but finitely many of the 2-cells are 2-spheres (i.e., have trivial attaching maps) so they contribute nothing significant to the resulting presentation ofG∼=π1(Z, z)∼=π1(Z, z).

Exercise

Prove that ifπ : A → C is a stack of CW complexes in which every fiber Fe is contractible, thenπis a homotopy equivalence. In fact,πis ahereditary homotopy equivalence, meaning that for every subcomplex D of C, π| : π⁻¹(D) → D is a homotopy equivalence. (Hint: The proof of the Rebuilding Lemma allows us to take A=C,π= idC andh=π.)

6.2 Decomposing groups which act on trees (Bass-Serre

6.2 Decomposing groups which act on trees (Bass-Serre Theory) 149 If eis an oriented 1-cell of a CW complex, we denote its initial and final points by o(e) andt(e) respectively (o for “origin” andt for “terminus”). A graph of groups is a system (G, Γ) consisting of: an oriented path connected graphΓ, a groupG(w) for each vertexwofΓ, a groupG(e) for each (oriented) 1-cell of Γ, and monomorphisms G(o(e)) ^φ

−

←−e G(e) ^φ

+

−→e G(t(e)). The group G(w) [resp.G(e)] is called avertex group [resp.edge group] of (G, Γ).

LetT be a maximal tree inΓ. As in Sect. 6.1, letE₀ [resp.E₁] be the set of vertices [resp. (oriented) 1-cells] of Γ. Associated with (G, Γ) and T is a group, denotedπ1(G, Γ;T), called thefundamental group of (G, Γ)based at T, namely: the quotient of the free product

'

w∈∗E₀G(w) (

∗F(E1) by the normal subgroup⁵generated by (i) elements of the forme⁻¹.φ⁻_e(x).e.(φ⁺_e(x))⁻¹where e∈E1 andx∈G(e), and (ii) elementse, whereeis a 1-cell of T.

This includes some well-known constructions as special cases:

Case 1: Free products with amalgamation. Here Γ =T =B¹, with two vertices, w(−) and w(+), and one 1-cell e. We are given monomorphisms φ^±_e :G(e)→G(w(±)). In this case

π1(G, Γ, T) =G(w(−)), G(w(+))|φ⁻_e(x)φ⁺_e(x)⁻¹∀x∈G(e). This is often written⁶ G(w(−)) ∗

G(e)

G(w(+)).

Case 2: HNN extensions. HereΓ = S¹, with one vertex w and one 1- cell e, T = {w}, and we are given monomorphisms φ^±_e : G(e) → G(w). In this case π1(G, Γ;T) = G(w), e | e⁻¹φ⁻_e(x)e(φ⁺_e(x))⁻¹ ∀x ∈ G(e). This is often written⁷ G(w)∗

φ where φ : φ⁺_e(G(e)) → φ⁻_e(G(e)) is the isomorphism φ⁻_e ◦(φ⁺_e)⁻¹.

Indeed, the general case ofπ1(G, Γ;T) is simply an iteration of these two special cases; for ifΓ =T thenπ1(G, Γ;T) is just a free product with an amal- gamation for each edge ofT; and in the general case (Γ =T),π1(G, Γ;T) is obtained from this “multiple” free-product-with-amalgamation by performing an HNN construction for each edge ofΓ which is not an edge of T.

Proposition 6.2.1. (Britton’s Lemma)For each w₀∈E₀, the homomor- phismγ_w0 :G(w₀)→π₁(G, Γ;T)induced byG(w₀)→

'

w∈∗E₀G(w) (

∗F(E₁) is a monomorphism.

5 As before,F(E₁) denotes the free group generated by the setE₁.

6 In general, ifG₁,G₂andAare groups andφi:AGiare monomorphisms, one writesG₁∗AG₂forG1, G₂|φ₁(a)φ₂(a)⁻¹∀a∈A; in this abuse of notation one is thinking ofφ₁(A),Aand φ₂(A) as the same group and one is “amalgamating G₁ andG₂ along the common subgroupA.”

7 In general, if G is a group, A and B are subgroups, and φ : A → B is an isomorphism, one writesG∗φforG, t|t⁻¹at=φ(a)∀a∈A. This group is the HNN extension ofGbyφ; the subgroupG(see 6.2.1) is thebase groupandtis called thestable letter. IfA=Gthis is anascending HNN extension.

Remarks on the proof. The version for Cases 1 and 2, above, is found in [106, Chap. IV, Sect. 2]. The general case then follows by the above remarks;

a direct proof is found in [142, p. 46]. All these involve reducing the length of a supposedly minimal non-trivial word in the kernel, thereby proving the kernel trivial. A topological proof is indicated in the next section; see Exercise 3 of Sect. 7.1.

Proposition 6.2.1 means that whenever we express a groupGnon-trivially as the fundamental group of a graph of groups, we are exhibiting a decompo- sition ofGinto “pieces” each of which is a subgroup ofG. Our purpose here is to give such a decomposition (Theorem 6.2.7) wheneverGacts rigidly on a tree.

We will need the “fundamental group of a CW complex based at a tree.”

Whenever ¯T is a tree in a CW complexX, the quotient map k:X →X/T¯ is a homotopy equivalence by 4.1.9. Let ¯v be the vertex of X/T¯ such that k( ¯T) = ¯v. We define the fundamental group of X based at T¯, de- noted π₁(X,T), to be the group¯ π₁(X/T ,¯ ¯v). For any vertex v of ¯T, k in- duces a canonical isomorphism k# : π1(X, v) → π1(X,T¯). The composition π1(X, v1)−→

k_1# π1(X,T¯)−→

k⁻¹

2#

π1(X, v2) is the isomorphismhτ of 3.1.11, where τ is any edge path in ¯T fromv1tov2. Note that if ¯T⁺is a maximal tree then π1(X,T¯) andπ1(X,T¯⁺) are canonically isomorphic.

A graph of pointed CW complexes is a system (X, Γ) consisting of: an oriented path connected graph Γ, a pointed path connected CW complex (X(w), x(w)) for each vertexwof Γ, a pointed path connected CW complex (X(e), x(e)) for each 1-celleofΓ, and cellular maps

(X(o(e)), x(o(e))) ^p

−

←−e (X(e), x(e)) ^p

+

−→e (X(t(e)), x(t(e))).

The total complex, Tot(X, Γ), is the adjunction complex obtained from {X(w) | w is a vertex of Γ} by adjoining

{X(e)×B¹ | e is an edge of Γ} exactly as in 6.1.3, using the maps p_e defined by: p_e(y,±1) = p^±_e(y) for all y ∈ X(e). The resulting map q : Tot(X, Γ) → Γ is a stack of CW complexes.

When each p^±_e# : π₁(X(e), x(e)) → π₁(X(w), x(w)) is a monomorphism, then (X, Γ) gives rise to a graph of groups (G, Γ), with vertex groups π1(X(w), x(w)), edge groupsπ1(X(e), x(e)), and monomorphisms p^±_e#.

There is a map s:Γ →Tot(X, Γ) such thats(w) =x(w) for each vertex w of Γ, and s maps each edge of Γ homeomorphically onto the image of {x(e)}×B¹in Tot(X, Γ). In fact,s(Γ) is a retract of Tot(X, Γ). The maximal treeT is mapped bysisomorphically onto a trees(T) in Tot(X, Γ).

Proposition 6.2.2.Let each p^±_e# be a monomorphism. There is an isomor- phismj making the following diagram commute for each vertexw ofΓ:

6.2 Decomposing groups which act on trees (Bass-Serre Theory) 151 π₁(Tot(X, Γ), x(w)) ←−−−−^βw π₁(X(w), x(w))

∼=

⏐⏐

^k# ⏐⏐^γw π1(Tot(X, Γ), s(T)) −−−−→_∼^j

= π1(G, Γ;T) whereβw is induced by inclusion, andγw is as in 6.2.1.

Proof. The tree s(T) can be extended, using 3.1.15, to a maximal tree T⁺ in Tot(X, Γ) such that, for each vertex w, T⁺∩X(w) is a maximal tree in X(w) (considered as a subcomplex of Tot(X, Γ)). The presentation of π1(Tot(X, Γ), x(w)) obtained by applying 3.1.16 to Tot(X, Γ), using T⁺, is seen by inspection to be a presentation of π1(G, Γ;T). In fact, the proof of 3.1.15 shows that an isomorphismjexists as indicated. This is clear when the complexes each have one vertex. But that is enough.

Now we are ready for our application of 6.1.3. LetY be a rigidG-tree.⁸This is a special case of what was considered in Sect. 6.1, so we carry over all the notation of that section except that we writeΓ in place ofV forG\Y, since it is a graph. In particular, whene∈E1, we havefe:Xe×{±1} →q⁻¹(Γ⁰) defined by the commutative diagram preceding Proposition 6.1.2. From that diagram one deduces several facts about the mapfe: (i)fe(Xe×{−1})⊂X_o(e). Define f_e⁻:Xe→X_o(e)byf_e⁻(y) =fe(y,−1). Then (ii)f_e⁻is a covering projection;

in fact, with obvious abuse of notation, it is the covering projectionG_e˜\X˜ → G_o(˜e)\X. Let˜ x_o(e) = r(˜x₀, o(˜e)). Then (iii) f_e⁻(x_e) = x_o(e). Similarly, (iv) we have a pointed covering projection f_e⁺ : (X_e, x_e) → (X_t(e), x_t(e)) where x_t(e)=r(˜x₀, t(˜e)).

Since x_o(e) andx_t(e) can be different fromxo(e) and xt(e), we do not yet have a graph of pointed CW complexes. We alter things to make one. For each e∈E₁, pick a cellular path α(e) in X_o(e) from x_o(e) to x_o(e), and a cellular pathβ(e) in X_t(e) from x_t(e) to x_t(e). Define H : (X_o(e)× {0})∪({x_o(e)} × I) → X_o(e) by (x,0) → x when x ∈ X_o(e), and by (x_o(e), λ) → α(e)(λ) when λ ∈ I. By 1.3.15, H extends to ¯H : X_o(e)×I → X_o(e). Let g_e⁻ : (X_o(e), x_o(e))→(X_o(e), x_o(e)) be the map g_e⁻(x) = ¯H(x,1). Similarly, define g_e⁺: (X_t(e), x_t(e))→(X_t(e), x_t(e)). Theseg_e^±are homotopic to the appropriate identity maps; by 1.3.16 we may assume they are cellular. On fundamental groups, we haveg⁻_e#=h_α(e)andg_e#⁺ =h_β(e)(in the notation of 3.1.11).

We now have a graph of pointed CW complexes (X, Γ) : (X(w), x(w)) is (X_w, x_w), (X(e), x(e)) is (X_e, x_e), p⁻_e : (X_e, x_e) →(X_o(e), x_o(e)) is the map g_e⁻◦f_e⁻, andp⁺_e =g_e⁺◦f_e⁺. The induced homomorphismsp^±_e# are monomor- phisms, by 3.4.6 and 3.1.11. Thus (X, Γ) induces a graph of groups (G, Γ) with G(w) = π1(Xw, xw), G(e) = π1(Xe, xe), φ⁻_e = p⁻_e# : π1(Xe, xe) →

8 In the literature one finds a “tree on whichGacts by simplicial automorphisms without inversions” a special case of a rigidG-tree.

π1(X_o(e), x_o(e)), andφ⁺_e =p⁺_e# : π1(Xe, xe) →π1(X_t(e), x_t(e)). By 6.2.2, the correspondingπ1(G, Γ;T) is isomorphic to π1(Tot(X, Γ), s(T)). Sinceg_e^± are homotopic to the identity maps, 4.1.7 and 6.1.3 imply thatπ1(Tot(X, Γ), s(T)) is isomorphic⁹ to π1(Z, z), whereZ and z =xv are as in Sect. 6.1. And we saw in that section thatπ1(Z, z) is isomorphic toG.

In summary, we have expressedGas the fundamental group of a graph of groups (G, Γ). But we are not quite done. We would like to identify the vertex and edge groups with specific subgroups¹⁰ ofG.

Observe that r⁻¹(Xo(e)) = ˜X ×p⁻¹(o(e)). Two of its path components are ˜X × {(o(e))^∼} and ˜X × {o(˜e)}. We have a covering projection r₁ :=

r|: ˜X × {(o(e))^∼} → X_o(e), and, using base points (˜v,(o(e))^∼) and x_o(e) = r(˜v,(o(e))^∼), we have an isomorphismχ1 :G_(o(e))∼ →π1(X_o(e), x_o(e)) as in 3.2.3. Similarly, r2 :=r|: ˜X × {o(˜e)} → Xo(e) is a covering projection, and we have an isomorphismχ2:Go(˜e)→π1(Xo(e), x_o(e)).

We picked a pathα(e) inX_o(e)fromx_o(e)tox_o(e). Let ˜α(e) be the lift of this path to ˜X×{(o(e))^∼}whose final point is (˜v,(o(e))^∼). Sincermaps the initial point of ˜α(e) to r(˜v, o(˜e)) =x_o(e), there must be a unique elementa(e)∈G such that the initial point of ˜α(e) is (a(e).˜v,(o(e))^∼). Thusa(e).o(˜e) = (o(e))^∼, and G_(o(e))∼ = a(e)G_o(˜e)a(e)⁻¹. Indeed, if a(e) ∈ G satisfies a(e).o(˜e) = (o(e))^∼, there is a path α(e) from x_o(e) to x_o(e) leading us to a(e) just as α(e) led us toa(e).

Letc_gdenote the conjugation isomorphismh→g⁻¹hg. One easily checks:

Proposition 6.2.3.The following diagram commutes:

G_o(˜e) ←−−−−^ca(e) G_(o(e))∼

⏐⏐

^χ2 ⏐⏐^χ1 π1(X_o(e), x_o(e)) −−−−→^hα(e) π1(X_o(e), x_o(e))

Similarly, corresponding to β(e) is an elementb(e) ∈ G and an isomor- phismcb(e):G(t(e))∼ →Gt(˜e)for which the corresponding diagram commutes.

Define a new graph of groups ( ¯G, Γ) by: ¯G(w) = Gw˜; ¯G(e) = Ge˜; ¯φ⁻_e is the compositionGe˜→G_o(˜e)

c⁻¹

−→a(e) G_(o(e))∼; and ¯φ⁺_e is the composition Ge˜→ G_t(˜e)

c⁻¹

−→b(e) G_(t(e))∼. The meaning of the next proposition will become clear in the proof:

Proposition 6.2.4.( ¯G, Γ)and(G, Γ) are isomorphic graphs of groups.

9 This is explained in more detail after 6.2.5, below.

10This is obviously desirable, and it will also enable us to avoid reliance on 6.2.1, which we have not yet proved.

6.2 Decomposing groups which act on trees (Bass-Serre Theory) 153 Proof. The following diagram commutes, as well as a similar diagram in which t(e) replaceso(e),b(e) replacesa(e), and + replaces−:

G_˜e  //

χ₃

G_o(˜e)

χ₂

c⁻¹

a(e) //G_(o(e))∼

χ₁

π1(Xe, xe)

f⁻

e# //π₁(X_o(e), x_o(e)) ^hα(e)//π₁(X_o(e), x_o(e))

Here, the right hand square comes from 6.2.3. The left hand square obviously

commutes.

It follows thatπ₁( ¯G, Γ;T) is isomorphic to G. But it does not follow that an isomorphism between those groups exists such that the canonical homo- morphismγ_w :G_w˜ →π₁( ¯G, Γ;T) can be identified with G_w˜ →G for all w.

To achieve this, we place restrictions on our choice of the cells ˜wand ˜e. (Until now, we have simply carried over the choices made in Sect. 6.1; each ˜w or ˜e was an arbitrary cell ofY “over”wor e.) We will need the following, which is proved using Zorn’s Lemma:

Proposition 6.2.5.There is a tree, T˜, inY such that pmaps T˜ isomorphi-

cally ontoT.

From now on, we choose each ˜wto be in ˜T, and each ˜eto be in ˜T whenever e is in T. The effect is that whenever e is a cell of T, o(˜e) = (o(e))^∼ and t(˜e) = (t(e))^∼. For those cells e,x_o(e) =x_o(e) andx_t(e)=x_t(e); and we pick α(e) and β(e) to be trivial, so thata(e) = 1 =b(e). Thus g_e^± = id whenever the edgeelies in T.

The subcomplex{x˜0} ×T˜⊂X˜×Y is a tree which contains all vertices of the form (˜x0,w). Let˜ T1=r({x˜0}×T);˜ T1is a tree inZcontaining all vertices xw; rmaps {˜x0} ×T˜ isomorphically ontoT1, andq maps T1 isomorphically ontoT.

By 4.1.8 and 6.1.3, there is a homotopy equivalenceh: Z →Tot(X, Γ).

To see this, apply 4.1.8 to the commutative diagram:

e∈E₁

Xe×B¹oo ? _

id

e∈E₁

Xe×S^{0 f} //

id

v∈E₀

Xv

id

e∈E₁

X_e×B¹oo ? _

e∈E₁

X_e×S^{0 g◦f} //

v∈E₀

X_v

wheref [resp.g◦f] agrees withf_e^± [resp.g^±_e ◦f_e^±] onX_e× {±1}. Indeed,h identifiesT₁ withs(T) in the natural way.

Now we can prove:

Proposition 6.2.6.There is an isomorphism ψ:π1( ¯G, Γ;T)→G such that for every vertex wof Γ the following diagram commutes (where γ¯w is analo- gous toγw₀ in 6.2.1):

Gw˜  //

¯ γ_w

G

π₁( ¯G, Γ;T)

∼

= ψ

::u

uu uu uu uu u

Proof. The requiredψis read off from the following commutative diagram:

Gw˜

χ₁

∼=

''O

OO OO OO OO OO O

¯ γ_w

 //G

∼= χ

π₁(X_w, x_w) ^β

w //

βR_wRRRRRRR((

RR RR R

γ_w

π₁(Z, x_w) ^k# //

∼= h_#

π₁(Z, T₁)

∼= h_#

vvmmmmmmmmmmmmm

π1(Tot(X, Γ), xw)

∼= k_#

π1(G, Γ;T)oo ^α

∼= π1(Tot(X, Γ), s(T))

π1( ¯G, Γ;T)

∼= χ

77o

oo oo oo oo oo

Here, χ is the isomorphism arising from 6.2.4;β_w is induced by inclusion.

To see thatψ is independent ofw, it is enough to check that the following diagram commutes, whenw₁ andw₂ are vertices ofΓ:

G ^χ

(1) //

χ⁽²⁾

π₁(Z, x_w1)

k⁽¹⁾

#

h_α#

xxppppppppppp

π₁(Z, x_w2)

k⁽²⁾

#

//π₁(Z, T₁)

Here,αis any path inT1fromxw₁toxw₂. We have seen thathα#= (k⁽²⁾_# )⁻¹◦ k⁽¹⁾_# , so we need only prove that the other triangle commutes. If g ∈ G, and χ⁽¹⁾(g) is represented by the loop ω in Z at x_w1, the lift, ˜ω, in ˜X×Y with initial point (˜x0,w˜1) has final point (g.˜x0, g.w˜1). The path αlies inT1; its lift ˜α with initial point (˜x0,w˜1) lies in {x˜0} ×T˜ and so has final point (˜x0,w˜2). Thus the lift of α⁻¹.ω.α with initial point (˜x0,w˜2) has final point (g˜x0, gw˜2). In other words,χ⁽²⁾(g) is represented byα⁻¹.ω.α. This means that

hα#◦χ⁽¹⁾(g) =χ⁽²⁾(g).

6.2 Decomposing groups which act on trees (Bass-Serre Theory) 155 Summarizing, we have shown how to decompose a group acting rigidly on a tree as the fundamental group of a graph of groups whose vertex groups and edge groups are stabilizers:

Theorem 6.2.7.Let G act rigidly on the tree Y with quotient p : Y → G\Y =: Γ. Let T be a maximal tree in Γ and let T˜ be a tree in Y such that p |: ˜T → T is an isomorphism. For each vertex w of Γ, let w˜ ∈ T˜ be such that p( ˜w) =w. For each 1-cell eof Γ pick a 1-cell e˜of Y mapped by p ontoe, subject to the rule that if e⊂T, thene˜⊂T˜. Orient each 1-cell e of Γ, and orient eache˜to make p|˜eorientation preserving. Leta(e)∈G[resp.

b(e)∈G] be such that a(e).o(˜e) = (o(e))^∼ [resp. b(e).t(˜e) = (t(e))^∼] subject to the rule that if e⊂T,a(e) =b(e) = 1. Let ( ¯G, Γ)be the graph of groups with vertex groups G_w˜, edge groups G_˜e, and monomorphisms

G_(o(e))∼ oo^c−1a(e) G_o(˜e)oo ? _G˜e  //G_t(˜e) ^c

−1

b(e) //G_(t(e))∼.

Then, letting ¯γ_w : G_w˜ → π₁( ¯G, Γ;T) denote the canonical homomorphism, there is an isomorphism ψ :π₁( ¯G, Γ;T)→G such that for each vertexw of Γ,ψ◦γ¯_w=inclusion:G_w˜→G. In particular, ¯γ_w is a monomorphism.

Remark 6.2.8.When the edgeeis not inT, it is customary to choose ˜esubject to the rule:o(˜e) = (o(e))^∼. Thena(e) = 1 for alle.

Example 6.2.9.The group SL2(Z) of 2×2 integer matrices of determinant 1 acts on the open upper half of the complex planeCby M¨obius transformations.

With respect to the hyperbolic metric ^ds_y this action is by isometries and its kernel is the subgroup{±I}. The orbit of{z|Im(z)>1}consists of that set together with a collection of pairwise disjoint open (Euclidean) circular disks whose closures inR²are tangent to the real axisRat rational points, one for each rational. Each such closure is also tangent to exactly two others (where we treat{z|Im(z)≥1}as the closure of a circular disk of infinite radius “tangent at ∞”). See Fig. 6.1. The complementary region is thus invariant under the action of SL2(Z) and contains an invariantSerre tree as illustrated in Fig.

6.1. This tree has vertices of order 2 and order 3 with respective stabilizers Z4 andZ6. Each edge contains one vertex of each kind and has stabilizer of order 2. With a little thought about how edge-stabilizers inject into vertex- stabilizers, one deduces from Theorem 6.2.7 thatSL2(Z) is isomorphic to the free product with amalgamationZ4∗_Z2Z6.

We now sketch the inverse construction: how to construct a rigid G-tree from a graph of groups.

Let (G, Γ) be a graph of groups, and let T be a maximal tree in Γ. Orient Γ. Form a graph of pointed CW complexes (X, Γ) by choosing pointed CW complexes (X(w), x(w)) and (X(e), x(e)), for every w and e, and pointed cellular maps p⁻_e : (X(e), x(e)) → (X(o(e)), x(o(e))) and

. . . . .

. . .

. . .

.

. . . .

. . .

. . .

. .

.

^x−axis

.

z = i

Schematic picture of the tree

Fig. 6.1.

p⁺_e : (X(e), x(e)) → (X(t(e)), x(t(e))) such that for suitable isomorphisms ψw : π1(X(w), x(w)) → G(w) and ψe : π1(X(e), x(e)) → G(e), p^±_e# can be identified with the monomorphismsφ^±_e in G. LetG=π1(G, Γ;T). By 6.2.2, Gcan be identified withπ1(Tot(X, Γ), x(v)) wherevis the base vertex ofΓ. Hence we have a free action ofGon U := (Tot(X, Γ))^∼. Applying Sect. 3.2, using 3.4.9 and 6.2.1, U is seen to be a quotient space obtained by gluing copies of (X(e))^∼×B¹ to copies of (X(w))^∼ via the lifts of the p’s, wheree andware variable. If we identify each copy of (X(w))^∼ inU to a point, and each copy of (X(e))^∼×B¹to a 1-cell, we obtain a graphY and a commutative diagram

U −−−−→^s Y

r

⏐⏐

⏐⏐^p Tot(X, Γ) −−−−→_q Γ

Here,ris the covering projection,sis the quotient map andpis a well defined map. Since s is easily seen to be a domination, and U is simply connected, Y is simply connected; i.e., Y is a tree, the Bass-Serre tree of (G, Γ). The freeG-action onU induces a rigid G-action onY;s is aG-map and pis the quotient map of theG-action. The treeT inΓ gives a trees(T) in Tot(X, Γ) as before; this lifts to a tree ˜s(T) inU which is mapped isomorphically to a tree ˜T inY. With choices as in Theorem 6.2.7, the resulting graph of groups ( ¯G, Γ) is isomorphic to the original (G, Γ).

Example 6.2.10.ABaumslag-Solitar group is a groupBS(m, n) with presen- tation x, t | t⁻¹x^mtx⁻ⁿ where m, n ≥ 1. Clearly, BS(m, n) is the HNN extensionZ∗φ_m,n whereφ_m,n:mZ→nZis the isomorphism taking mto n.

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].