The goal of this section is Theorem 13.21, the main theorem of this thesis, which shows that the centraliser of every higher Dehn twist automorphism in Out(Fn) or Aut(Fn) satisfies finiteness property VF.
Definition 13.15. A groupGhasfiniteness property Fif there is a finite CW complex which is aK(G,1)-space. Ghasfiniteness property VFif it has a subgroupH of finite index which has property F.
We first need some elementary preparation.
Lemma 13.16.
(i) If G satisfies property F, and G0 is a subgroup of finite index in G, then G0 satisfies property F.
(ii) If G0 is a subgroup of G of finite index, then G satisfies VF if and only if G0 does.
(iii) If G1, . . . , Gl satisfy F (or VF respectively), then so doesLl i=1Gi. (iv) All finitely generated abelian groups have finiteness property VF.
Proof. In (i), a finite K(G0,1) space can be obtained as a covering space of a finite K(G,1) space with finitely many sheets.
In (ii), ifH0 has finite index inG0 and property F, it also has finite index inG, soG has property VF. Conversely, ifH has finite index in Gand satisfies property F, then H0 :=H∩G0 has finite index inG0. AsH0 also has finite index inH, part (i) shows thatH0 has property F, so G0 satisfies VF. This proves (ii).
If each Xi is a finite K(Gi,1) space in (iii), then Q
iXi is a finite K(G,1) space.
This proves (iii) for property F. For property VF, we use thatL
Hi has finite index in LGi when eachHi has finite index inGi.
As every finitely generated abelian group is a finite direct sum of cyclic groups, (iii) reduces (iv) to the special case of cyclic groups. Using (ii), we only have to show finiteness property VF for the trivial group and for Z. But these groups have finite classifying spaces, namely a point and a circle respectively.
Proposition 13.17. If 1 → G0 → G→ G00 → 1 is a short exact sequence of groups, G0 has finiteness property F, andG00 has finiteness property VF, thenG has finiteness property VF.
Proof. Let H00 be a finite index subgroup of G00 satisfying finiteness property F, and denote its preimage inG byH. We then have a short exact sequence
1→G0→H →H00→1.
By Theorem 7.1.10 in [16], we can construct a finite classifying space forH from finite classifying spaces forG0 andH00. AsHhas finite index inG, the groupGhas property VF.
Proposition 13.18. Suppose that in the short exact sequence1→G0 →ι G→π G00→1, the group G0 has finiteness property VF, and G00 is finitely generated abelian. Then G has finiteness property VF.
Proof. Suppose first that G00 ∼=Z. Let H0 be a subgroup of some finite index din G0 such thatH0 satisfies property F. Then the intersectionK0 of all (finitely many) index dsubgroups ofG0 also satisfies property F by Lemma 13.16(i). Furthermore, the group K0 is a characteristic subgroup ofG0, i.e. it is left invariant by every automorphism of
G0. Let now t ∈G be an element such that π(t) generates G00. We denote by K the subgroup ofG generated by tand ι(K0). It has finite index inG and fits into a short exact sequence
1→K0 →K→G00→1.
Proposition 13.17 implies thatKand henceGhas property VF. This finishes the proof forG00∼=Z.
By induction on m, we now prove the assertion in the case that G00 ∼= Zm. Let A00∼=Zbe a direct summand inG00, soG00/A00∼=Zm−1. The short exact sequences
1→G0 →π−1(A00)→A00→1, 1→π−1(A00)→G→G00/A00→1,
and the induction hypothesis now prove the assertion for free abelianG00.
We now come to the general case. Let H00 be a free abelian subgroup ofG00 of finite index and H=π−1(H00)⊂G. We obtain a short exact sequence
1→G0 →H→H00→1.
Hence H satisfies VF, and Lemma 13.16(ii) implies VF forGas well.
Proposition 13.19. If C is a finite set of conjugacy classes in Fn, then Aut(Fn,C) and Out(Fn,C) satisfy finiteness property VF.
Proof. This is shown in Corollary 6.1.4 of [14] for Out(Fn,C). As Inn(Fn) ∼= Fn (or Inn(F1) = 1) has propertyF, the exact sequence
1→Inn(Fn)→Aut(Fn,C)→Out(Fn,C)→1 and Proposition 13.17 show VF for Aut(Fn,C) as well.
Theorem 13.20. All relative centralisers have finiteness property VF.
Proof. We show by induction on d:= deg(G) that SI0(D∗v,(ηi)) and SI0(D,b (ηi)) have finiteness property VF whenever Dis (a truncatable replacement of) a prenormalised higher Dehn twist on G.
Let first d≤ 1. The homomorphisms A∗v and Ab in Proposition 13.13 have kernels which are direct products of finitely many free groups, hence ker(A∗v) and ker(A)b satisfy property F. The images of A∗v and Ab have finite index in the direct sum of finitely many groups of the form Out(Gw,Cw) or Aut(Gv,Cv). By Proposition 13.19 and Lemma 13.16, we conclude that the images ofA∗v andAbhave finiteness property VF. Proposition 13.17 shows that SI2(D∗v,(ηi)) and SI2(D,b (ηi)) satisfy VF whenever (D,I,(ηi)) satisfies hypothesis (S) of Definition 13.5. In the following, we exhibit the arguments forSI2(D,b (ηi)), which we simply denote bySI2. The proof works in the same way forSI2(D∗v,(ηi)).
Recall that CI2 is defined to be the kernel of evh :CI1 →P in (78) on page 125, and SI2 =CI2∩SI0 is the kernel of the restricted homomorphism evh:SI1 →P. The groupP
is finitely generated abelian. As any subgroup of a finitely generated abelian group is finitely generated abelian, evh(S1I)⊂P is finitely generated abelian. Proposition 13.18 shows thatSI1 has property VF.
The target group of the rotation homomorphism evr in (77) on page 125 is finitely generated abelian, so the image of evr is. Proposition 13.18 shows thatSI0 satisfies VF.
By Lemma 13.6 these groups SI0 also satisfy VF if we do not require hypothesis (S).
This finishes the proof ford≤1.
We now prove inductively the assertion for d ≥ 2. Suppose first that hypothesis (S) is satisfied for SI1(D,b (ηi)) or SI1(D∗v,(ηi)). By Proposition 13.14 this group is a direct sum of finitely many relative centralisers of degree d−1. By induction and Lemma 13.16(iii), we conclude that SI1(D∗v,(ηi)) and SI1(D,b (ηi)) have property VF.
As this is the kernel of evr in (77), and the image of that homomorphism is finitely generated abelian, Proposition 13.18 shows thatSI0 has property VF. By Lemma 13.6 this also holds true without assuming (S).
We are now in a position to deduce the main theorem of this thesis.
Theorem 13.21. Whenever D is a higher Dehn twist on a higher graph of groups G with finitely generated free fundamental group, then the centralisersC(D∗v) andC(D)b satisfy property VF.
Proof. If we have Db = ρbDc0ρb−1 for two higher Dehn twists D ∈ Aut0(G) and D0 ∈ Aut0(G0), then C(D)b ∼= C(Dc0). Therefore Theorem 8.6(ii) reduces the statement to normalisedDforC(D). Similarly, we can reduce to pointedly normalisedb DforC(D∗v) by Theorem 8.6(i).
Note that C0(D) is the relative centraliser withb I =∅and the empty partition I in Definition 13.12. Therefore Theorem 13.20 shows that it satisfies VF. By Lemma 13.1, it has finite index inC(D), so Lemma 13.16(ii) shows thatb C(D) satisfies property VF.b Similar arguments apply toC(D∗v).
Remark 13.22. When we have a finite set C of conjugacy classes in π1(G, v), then we can also look at the groupGof those (H,(δi)i∈I)∈SI0(D,(ηi)) such thatH∗ fixes each class in C. A refined version of the lemmas in this chapter can be used to show that groups of the formG/(G∩KAI) or G/(G∩KOI) satisfy property VF whenD is (a truncatable replacement of) a prenormalised higher Dehn twist. This specialises to Theorem 13.20 whenC=∅.
When I =∅, then we obtain that the intersections C(D∗v)∩Aut(π1(G, v),C) and C(D)b ∩Out(π1(G, v),C) satisfy VF for every higher Dehn twist D on G with finitely generated free fundamental group.
14 Isometric CAT(0) actions
In this chapter we discuss the translation length of an element g in a group G acting by isometries on a CAT(0) space. We first recall some definitions.
A CAT(0) space is a geodesic metric spaceX satisfying theCAT(0)-inequalityfor all p, q, r∈X: That is, for all geodesic triangles with vertices p, q, r∈X and all x∈[p, q]
andy∈[p, r], the euclidean comparison triangle withd(¯p,q) =¯ d(p, q),d(¯p,r) =¯ d(p, r), d(¯q,r) =¯ d(q, r), d(¯p,x) =¯ d(p, x), d(¯p,y) =¯ d(p, y), ¯x ∈ [¯p,q], ¯¯ y ∈ [¯p,r] satisfies¯ d(x, y)≤d(¯x,y). The situation is depicted in Figure 10.¯
X R2
p q
r
x y
¯
p q¯
¯ r
¯ x
¯ y
Figure 10: The CAT(0) inequality.
Taking q =r, it follows that there is a unique geodesic between p and q, which we denote by [p, q].
A more precise definition of CAT(0) spaces can be found in Chapter II.1 of [11].
Recall that X is called complete if every Cauchy sequence converges. The space X is called properif all closed ballsB(x, r) are compact.
14.1 Translation lengths and centralisers
Definition 14.1. Letγ :X →X be an isometry. Itstranslation length is
|γ|:= inf
x∈Xd(x, γ(x)).
γ is called
• elliptic, if |γ|= 0 and the infimum is attained at some x ∈ X, i.e. x is a fixed point ofγ,
• hyperbolic, if|γ|>0 and the infimum is attained at some x∈X,
• parabolic, if the infimum is not attained,
• semisimple, if it is either elliptic or hyperbolic.
Translation lengths are invariant under conjugation, i.e. |α|= |βαβ−1|. Moreover, α is elliptic, parabolic, or hyperbolic respectively, if and only if βαβ−1 is. Similarly
|α|=|α−1|.
If Q is any group and g ∈Q, then we denote by JgK the class in the abelianisation H1(Q) =Q/[Q, Q] represented byg.
There is the following criterion for translation lengths using centralisers: Suppose g ∈ G has positive translation length for an action on a proper CAT(0) space X.
Then it can be shown that it fixes a point in the boundary of X. The action of the centraliserC(g) preserves the set of horospheres at this point, but it might shift them by a certain length. This gives rise to a group homomorphismC(g)→Rsending g to the translation length|g|>0. Since Ris abelian and torsion-free, this is possible only if JgK ∈ H1(C(g)) has infinite order. This gives rise to the following theorem, which has also appeared implicitly in the proof of Theorem 1 in [9].
Theorem 14.2(Bridson, Karlsson, Margulis). LetGbe any group andg∈G. Assume thatJgK has finite order in H1(C(g)). Then |g|= 0 whenever G acts by isometries on a proper CAT(0) space.