Topological Finiteness Properties and Dimension of Groups
7.2 Finiteness properties and dimensions of groups
LetGbe a group. Even thoughK(G,1)-complexes always exist and are unique up to homotopy equivalence, there may or may not existK(G,1)-complexes having special properties. For example, we say that G has type Fn if there exists aK(G,1)-complex having finite n-skeleton.
Proposition 7.2.1.Every group has type F0;Ghas type F1 iffGis finitely generated; Ghas typeF2 iffGis finitely presented; for n≥2,Ghas typeFn
iff there exists a finite pointedn-dimensional (n−1)-aspherical CW complex (X, x)such thatπ1(X, x)is isomorphic to G.
Proof. Every group has typeF0, by 7.1.5. LetGhave typeFnand let (Z, z) be aK(G,1)-complex with finiten-skeleton. Forn= 1 [resp.n= 2],Gis finitely generated [resp. finitely presented], by 3.1.17. For n ≥2, π1(Zn, z)∼= Gby 3.1.7;Zn is the required (n−1)-aspherical complex.
Conversely, if G is finitely generated [resp. finitely presented], we saw in 1.2.17 and 3.1.9 how to build a 2-dimensional CW complexXwith one vertex xandπ1(X, x)∼=G, so thatX1 [resp.X] is finite. Attach cells of dimension
≥3 as in the proof of 7.1.5 to get the requiredK(G,1) having finite 1-skeleton [resp. 2-skeleton]. Ifn≥2 and (X, x) is a finite (n−1)-aspherical CW complex such thatπ1(X, x)∼=G, attach cells of dimension≥n+ 1 as in 7.1.5 to build
aK(G,1) having finite n-skeleton.
Here is a restatement of 7.2.1:Ghas typeFniff there is an(n−1)-connected free n-dimensional G-CWcomplex which is finite mod G. This is clear when n≥2 and is an exercise whenn= 1.
For everyn, there are groups of typeFn which do not have typeFn+1. An efficient method of constructing such groups is given in Sect. 8.3. Such groups
170 7 Topological Finiteness Properties and Dimension of Groups
also occur “in nature,” for example in certain families ofS-arithmetic groups, see [1].
A CW complex each of whose skeleta is finite hasfinite type. We say that Ghastype F∞ if there exists aK(G,1)-complex of finite type.
Proposition 7.2.2.Ghas typeF∞ iffGhas typeFn for all n.
Proof. “Only if” is trivial. To prove “if,” apply 7.1.13 infinitely often, by induction onn, gradually building aK(G,1)-complex of finite type.
Next, we show that the propertiesFn are invariant under passage to and from a subgroup of finite index.
Proposition 7.2.3.Let H≤Gand let[G:H]<∞. LetGandH have type Fn−1. Then Ghas typeFn iffH has typeFn.
Proof. “Only if” follows from 3.2.13, 7.1.4 and 7.2.1. When n = 1 [resp.
n = 2], “if” is the well-known elementary fact that a group containing a finitely generated [resp. finitely presented] subgroup of finite index is itself finitely generated [resp. finitely presented]. It remains to prove “if” when n≥3.
LetX be aK(G,1)-complex having finite (n−1)-skeleton. Letp: ¯X→X be a finite-to-one covering projection, where ¯X is a K(H,1)-complex: see 3.2.13 and 7.1.4. Note that ¯Xhas finite (n−1)-skeleton. Sincen≥3 andHhas typeFn, Theorem 7.1.13 gives aK(H,1)-complex ¯Y having finite n-skeleton such that ¯Yn−1= ¯Xn−1. Thus ¯Ynis obtained from ¯Xn−1by attaching finitely manyn-cells, and ¯Yn is (n−1)-aspherical. Let{fα:Sn−1→X¯n−1|α∈ A}
be attaching maps for these n-cells, A being finite. Attach n-cells to Xn−1 by the maps p◦fα : Sn−1 → Xn−1. Let the resulting n-dimensional CW complex be Zn. Let ¯Zn be the covering space of Zn corresponding to the subgroup H (appropriate base points being understood everywhere). Then Y¯n is a subcomplex of ¯Zn, and ¯Yn−1= ¯Zn−1= ¯Xn−1. Since ¯Yn is (n−1)- aspherical, so is ¯Zn, hence also Zn, by 7.1.4. Thus Ghas typeFn, by 7.2.1.
Corollary 7.2.4.Let H ≤G and let [G:H]<∞. For 0 ≤n≤ ∞,G has typeFn iffH has typeFn.
Proof. Apply 7.2.3 inductively. For the casen=∞then apply 7.2.2.
The one-point space is a K(G,1)-complex where G is the trivial group.
Since the trivial group has finite index in every finite group, we have:
Corollary 7.2.5.Every finite group has typeF∞. Among the notable groups of typeF∞ is Thompson’s groupF, discussed in later chapters. Many torsion free groups of type F∞ turn out to have
a stronger property, type F, which is discussed below. (The group F is an exception to this.6)
We now turn to the question of minimizing the dimension of a K(G,1)- complex. Thegeometric dimension ofGis∞if there does not exist a finite- dimensional K(G,1)-complex; otherwise it is the least integer d for which there exists ad-dimensional K(G,1)-complex.
Proposition 7.2.6.Ghas geometric dimension 0iffGis trivial.Ghas geo- metric dimension1iffGis free and non-trivial. IfGhas geometric dimension d, every subgroup ofGhas geometric dimension≤d.
Proof. The dimension 0 statement is clear. By 3.1.16, every 1-dimensional CW complex has free fundamental group, so ifGhas geometric dimension 1, Gis free and non-trivial. Conversely, if G is free and non-trivial, we saw in Example 1.2.17 how to build a 1-dimensional CW complex whose fundamental group isG(by 3.1.8). Its universal cover is contractible, by 3.1.12, so it is a 1-dimensionalK(G,1).
Finally, let X be a d-dimensional K(G,1)-complex, and let H ≤ G. By 3.2.11,X has a covering space ¯X whose fundamental group isH. By 7.1.4, ¯X
is ad-dimensionalK(H,1)-complex.
There is no question of replacing the inequality by equality in the last part of 7.2.6; just consider the trivial subgroup of the free groupZ. Even for subgroups of finite index there are limitations: we will see in 8.1.5 that a finite cyclic group has infinite geometric dimension, while its trivial subgroup (of finite index) has geometric dimension 0. Nevertheless we have:
Theorem 7.2.7. (Serre’s Theorem)Let Gbe torsion free, and let H be a subgroup of finite index having finite geometric dimension. ThenGhas finite geometric dimension.
Proof. LetY be a finite-dimensionalK(H,1)-complex and letH¯g1, . . . , H¯gn
be the cosets ofH in G. Let ˜Y be the universal cover of Y. Let ˜X = n i=1
Y˜i
where each ˜Yi= ˜Y. Then ˜X is a finite-dimensional contractible CW complex.
We describe a (left)G-action on ˜X. We have selected the coset representatives
¯
g1, . . . ,¯gn. A right action ofGon the set{1,· · · , n}is defined by the formula (i, g) → i.g where ¯gig ∈ Hg¯i.g. Indeed, we can write ¯gig =h(g, i)¯gi.g, thus associating withg∈Gann-tuple (h(g,1),· · · , h(g, n)) of elements ofH. The required leftG-action on ˜X is
g.(y1,· · ·, yn) = (h(g,1)y1.g,· · ·, h(g, n)yn.g).
This action ofGclearly makes ˜X into a rigidG-CW complex. It remains to prove that the action is free.
6 This is a place where the two uses of the letterF might cause confusion: “type F” and “the groupF”.
172 7 Topological Finiteness Properties and Dimension of Groups
Let g.(y1,· · · , yn) = (y1,· · · , yn). Letm be such that i.gm =i for all i;
sinceg permutes a finite set such anmexists. Then
gm.(y1,· · ·, yn) = (h(gm,1)y1,· · ·, h(gm, n)yn) = (y1,· · · , yn).
So each h(gm, i) = 1 because H acts freely on ˜Y. So ¯gigm = ¯gi for all i, implyinggm= 1. Thusg= 1 sinceGis torsion free.
Remark 7.2.8.TheG-action on n i=1
Y˜i described in this proof does not restrict to the diagonalH-action.
One way of showing that a groupGhas geometric dimension≤dis to find some contractibled-dimensional freeG-CW complex, since by 3.2.1 and 7.1.3 the quotient complex will be a d-dimensional K(G,1)-complex. One way of showing thatGhas geometric dimension≥dis to show thatHd(X;R)= 0 for some K(G,1)-complexX and some ring R, applying 7.1.7, 2.5.4 and 2.4.10.
For example, recall that thed-torusTdis thed-fold product of copies ofS1. As explained in Sect. 3.4, ˜Tdis homeomorphic toRd, soTdis aK(Zd,1)-complex.
Proposition 7.2.9.Hd(Td;Z)∼=Z.
Proof. GiveRthe CW complex structure with vertex set Zand with 1-cells [m, m+1] for eachm∈Z. GiveRnthe product structure and regardTnas the quotient complex ofRn by the obvious free action of Zn onRn (translation by integers in each coordinate); see 3.2.1. ThenTnhas just onen-cell,en.
Orient the 0-cells ofRby +1. Orient [m, m+ 1] by the characteristic map I1→[m, m+ 1],t→ 12(t+ 2m+ 1). GiveRn the product orientation. Then theZn-action is orientation preserving and the quotientq : Rn → Tn gives an orientation toTn. It is enough to prove thaten∈Cn(Tn;Z) is a cycle.
Let ˜en = [0,1]n ∈ Cn(Rn;Z). For 1 ≤ i ≤ n and = 0 or 1, let ˜eni,−1 = [0,1]i−1 × {} × [0,1]n−i. By 2.5.17, [˜en : ˜eni,0−1] = (−1)i and [˜en : ˜eni,1−1] = (−1)i+1. Clearly en = q#(˜en). Let hi : Rn → Rn be the translation (x1, . . . , xn) → (x1, . . . , xi−1, xi+ 1, xi+1, . . . , xn). Then hi(˜eni,0−1) = ˜eni,1−1. Moreover, as chains, hi#(˜eni,0−1) = ˜eni,1−1. Since q◦hi = q, q#(˜eni,1−1) =q#◦hi#(˜eni,0−1) =q#(˜eni,0−1).
∂en=∂(q#(˜en))
=q#(∂˜en), by 2.4.3
=q#
n
i=1
(−1)i(˜eni,0−1−e˜ni,1−1)
by 2.5.17
= n i=1
(−1)i(q#(˜eni,0−1)−q#(˜eni,1−1))
= 0.
Corollary 7.2.10.Zd has geometric dimensiond.
Combining this with 7.2.6 gives:
Corollary 7.2.11.If G has a free abelian subgroup of rank d, then G has
geometric dimension≥d.
In particular, 7.2.11 can be useful for showing thatGhas infinite geometric dimension.
We have mentioned (and we will prove in 8.1.5) that every non-trivial finite cyclic group has infinite geometric dimension. Hence, by 7.2.6:
Proposition 7.2.12.Every group containing a non-trivial element of finite
order has infinite geometric dimension.
A groupGhastype F if there exists a finiteK(G,1)-complex. Groups of typeF discussed in this book include: finitely generated free groups, finitely generated free abelian groups, and torsion free subgroups of finite index in finitely generated Coxeter groups. Other important examples are: torsion free subgroups of finite index in arithmetic groups: see [29]; and torsion free sub- groups of finite index in the outer automorphism group of a finitely generated free group: see [44].
IfGhas typeF thenGhas typeF∞andGhas finite geometric dimension.
It is natural to ask if the converse is true. We say thatGhastypeF Dif some (equivalently, any)K(G,1)-complex is finitely dominated.
Proposition 7.2.13.G has type F D iff G has type F∞ and G has finite geometric dimension.
Proof. “If”: Let X be a K(G,1)-complex of finite type and let Y be a d- dimensionalK(G,1)-complex. By 7.1.7 there are mapsY −→f X −→g Y such that g◦ f idY. By 1.4.3, we may assume f and g are cellular, so that f(Y)⊂Xd. There are induced maps Y −→f Xd −→g| Y whose composition is homotopic to idY; andXd is finite.
“Only if”: Let X be aK(G,1)-complex and let X −→f Y −→g X be ho- motopic to idX, where Y is a finite CW complex. By 4.3.5, X is homotopy equivalent to Tel(f◦g), which is finite-dimensional, soGhas finite geometric dimension. To show thatGis of typeF∞we will show by induction thatGis of typeFnfor alln. By 7.2.2 this is enough. CertainlyGis of typeF0. AssumeG is of typeFn−1. LetXbe aK(G,1)-complex such thatXn−1is finite andX is dominated by a finite complex. Then there is a finite subcomplexKofX and a homotopyD:X×I→Xsuch thatD0= idX andD1(X)⊂K. LetLbe a finite subcomplex7ofX such thatD((Xn−1∪K)×I)⊂L. We claim (X, L) is n-connected. To see this, let φ : (Bk, Sk−1) → (X, L) be a cellular map
7 We are using 1.2.13 and 1.4.3 repeatedly.
174 7 Topological Finiteness Properties and Dimension of Groups
wherek≤n. Thenφ(Sk−1)⊂Xn−1, soDt◦φ(Sk−1)⊂Lfor allt∈I. More- over,D1◦φ(Bk)⊂K⊂L. By 1.3.9, there is a strong deformation retraction F :Bk×I×I→Bk×I, ofBk×Ionto (Bk×{1})∪(Sk−1×I). The required ho- motopyΦ:Bk×I→X, relSk−1, ofφintoLisΦ(s, t) =D◦(φ×id)◦Ft(s,0).
By 4.2.1,X is homotopy equivalent to a CW complexY such that Yn =Ln
is finite. ThusGhas typeFn.
Corollary 7.2.14.Let Gbe a group and let H be a subgroup of finite index.
IfGhas typeF D, so hasH. IfH has typeF Dand ifGis torsion free, then
Ghas typeF D.
Note that the trivial group{1} has typeF D(indeed, typeF) while non- trivial finite groups, in all of which{1}has finite index, do not have typeF D, by 7.2.12.
The question remains: does typeF Dimply typeF? The general question of when a finitely dominated CW complex X is homotopy equivalent to a finite CW complex is understood: the only obstruction (Wall’s finiteness ob- struction) lies in the reduced projective class group ˜K0(Z[π1(X)]). See, for example, [29, Chap. VIII, Sect. 6]. Non-trivial obstructions occur, but it is unknown at time of writing whether the obstruction can be non-zero whenX is aspherical.
The analog of 7.2.14 for typeF is also unknown: obviously if [G:H]<∞ and ifGhas typeF thenH has typeF (by 7.1.4 and 3.2.13). But for torsion freeGthe converse is unknown.
Proposition 7.2.15.If Ghas typeF D, thenG×Z has typeF.
Proof. This follows from 4.3.7. In detail, letX −→f Y −→g X be cellular maps, where X is aK(G,1)-complex, Y is a finite CW complex, and g◦f idX. ThenX×S1is aK(G×Z,1)-complex which, by 4.3.5, is homotopy equivalent
to the finite mapping torusT(f◦g).
Corollary 7.2.16.If G has type F D, then G is a retract of a group G of type F; i.e., there are homomorphisms G → G → G whose composition is
idG, where the first arrow is an inclusion.
Proposition 7.2.17.If there is aK(G,1)-complex which is dominated by a d-dimensional CW complex thenG×Zhas geometric dimension≤d+ 1.
Proof. Let Y dominate X, where X is a K(G,1)-complex and Y is d- dimensional. As in the proof of 7.2.15, X ×S1 is homotopy equivalent to a (d+ 1)-dimensional CW complex, which is therefore aK(G×Z,1)-complex.
Corollary 7.2.18.If there is a K(G,1)-complex which is dominated by ad- dimensional CW complex thenGhas geometric dimension≤d+ 1.
Remark 7.2.19.The conclusion of 7.2.18 can be improved to “Ghas geometric dimension≤d,” except possibly whend= 2, where the situation is not yet understood. The proof of this can be found, for example, in [29, Chap. VIII, Sect. 7]. This proof is accessible to readers of the present chapter and is only omitted to save space. Note that it uses the Relative Hurewicz Theorem 4.5.1.
Here is a useful necessary and sufficient condition for typeFn+1:
Theorem 7.2.20.Let n ≥1, let the group Ghave type Fn, and let X be a K(G,1)-complex with finite n-skeleton. Then Ghas type Fn+1 iff there is a K(G,1)-complex Y with finite(n+ 1)-skeleton and Yn=Xn.
Proof. “If” is clear. “Only if” follows from 7.1.13 whenn≥2 and is obvious
whenn= 1.
In the next proof we suppress base points in homotopy groups to simplify notation:
Theorem 7.2.21.LetN GQbe an exact sequence of groups. IfGhas typeFn and if N has typeFn−1 thenQhas type Fn.
Proof. This is obvious forn ≤2 so we assume n ≥3. LetY be an (n−2)- aspherical finite (n−1)-dimensional CW complex whose fundamental group is isomorphic toQ, and let X be a K(G,1)-complex. As before, we consider the commutative diagram
X˜ ×Y˜ projection //
r
Y˜
p
Z q //Y
where r is the covering projection obtained from the diagonal action of G on ˜X ×Y˜ (G acts on ˜Y via Q). Then q is a fiber bundle whose fiber is theK(N,1)-complexN\X. It follows from the exact sequence in 4.4.11 that˜ q# : πn−1(Z)→ πn−1(Y) is an epimorphism. The map q is also a stack of CW complexes. SinceNhas typeFn−1, there is aK(N,1)-complexW having finite (n−1)-skeleton, which is of course homotopy equivalent toN\X˜. By the Rebuilding Lemma 6.1.4 there is a diagram
Z
q
A
AA AA AA A
h
Y
Z
q
>>
}} }} }} }
176 7 Topological Finiteness Properties and Dimension of Groups
which commutes up to homotopy, whereq is a stack of CW complexes with fibers W, h is a homotopy equivalence, and Z has finite (n−1)-skeleton.
SinceZn−1is (n−2)-aspherical with fundamental group isomorphic toG, the same is true of (Z)n−1. Since G has typeFn, 7.2.20 implies that there is a K(G,1)-complexX with finiten-skeleton whose (n−1)-skeleton is (Z)n−1. The inclusion map (Z)n−1→Z induces an epimorphism onπn−1, hence we can attach finitely many n-cells to Z to kill πn−1(Z). Since q induces an epimorphism onπn−1, the same is true ofY. Thus, using 7.2.20 again, we see
thatQhas typeFn.
This theorem should be compared with Exercise 1 where it is asserted that ifN andQhave typeFn thenGhas typeFn.
Exercises
1. LetNGQbe a short exact sequence of groups. Prove that ifN has type FnandQhas typeFn thenGhas typeFn.Hint: see Theorem 7.1.10.
2. Devise similar exercises involving finite geometric dimension, typeF Dand type F.
3. Prove that if G is the fundamental group of a finite graph of groups whose vertex groups have typeFnand whose edge groups have typeFn−1thenGhas typeFn.Hint: Use 7.1.9 and 6.1.4.
4. How many other proofs of 7.2.9 can you find?
5. Sharpen 7.2.13 by specifying the dimensions: (i) “dominated by a finite d- dimensional complex” implies “geometric dimension ≤ d+ 1”; (ii) “Fd and geometric dimension ≤d” implies “dominated by a finited-dimensional com- plex.”
6. Give an example of a short exact sequence of groups N G Q where G andQhave typeF, andNis not finitely generated. (Thus one cannot expect a theorem in the spirit of 7.2.21 and Exercise 1 of Sect. 7.2 for this case.)