ON AUTOMORPHISM GROUPS OF FREE PRODUCTS OF FINITE GROUPS, I: PROPER ACTIONS
YUQING CHEN, HENRY H. GLOVER, AND CRAIG A. JENSEN
Abstract. If G is a free product of finite groups, let ΣAut1(G) denote all (necessarily symmetric) automorphisms of G that do not permute factors in the free product. We show that a McCullough-Miller [D. McCullough and A. Miller, Symmetric Automorphisms of Free Products, Mem. Amer. Math. Soc. 122 (1996), no. 582] and Guti´errez-Krsti´c[M. Guti´errezand S. Krsti´c, Normal forms for the group of basis-conjugating automorphisms of a free group, Inter- national Journal of Algebra and Computation 8 (1998) 631-669] derived (also see Bogley-Krsti´c[W. Bogley and S. Krsti´c, String groups and other subgroups of Aut(Fn), preprint] space of pointed trees is an EΣAut1(G)-space for these groups.
1. Introduction We remind the reader (see [8, 11]) that if G is a discrete group, then the con- tractible G-space EG is characterized (up to G-equivariant homotopy) by the prop- erty that if H is any subgroup of G then the fixed point subcomplex EGH is con- tractible if H is finite and empty if H is infinite. These spaces are basic tools in studying the geometry of the group G. Let G be a free product of n finite groups. We wish to construct an EΣAut1(G)- space based on McCullough-Miller’s [10] space of trees, which uses rooted trees similar to those found in Guti´errez-Krsti´c[6]. Here ΣAut1(G) is the kernel of the projection ΣAut(G) Σn. We will show: → Theorem 1.1. Let G be a free product of finite groups and let ΣAut1(G) = Aut1(G) denote all automorphisms of G that do not permute groups in the free product. The space L(G) = EΣAut1(G). That is, L(G) is a contractible space which ΣAut1(G) acts on with finite stabilizers and finite quotient. Moreover, if F is a finite subgroup F of ΣAut1(G), then the fixed point subcomplex L(G) is contractible. We pause to note a few other related papers. In [4] Collins and Zieschang establish the peak reduction methods that underly all of the contractibility argu- ments here. Gilbert [5] further refines these methods and gives a presentation for ΣAut(G). In [3] Bridson and Miller show that every finite subgroup of ΣAut1(G) fixes a point of McCullough-Miller space K0(G). In [1] Bogley and Krsti´ccom- pletely calculate the cohomology of ΣAut(Fn). Brady, McCammond, Meier, and Miller [2] use McCullough-Miller space to show that ΣAut(Fn) is a duality group. The paper is structured as follows. In section 2, we define the space L(G) and in section 3 we show it is contractible using a standard norm. In section 4 we briefly
Date: November 5, 2002. 1991 Mathematics Subject Classification. 20E36 (20J05). Key words and phrases. automorphism groups, free products, proper actions. 1 2 examine stabilizers of points in L(G). In section 5, we develop many new norms, each of which can be used to show L(G) is contractible. In section 6, we classify F fixed point subcomplexes L(G) where F is a finite subgroup of ΣAut1(G) and in section 7 we show that these subcomplexes are contractible. Section 8 contains a counterexample showing that if F is a finite subgroup of ΣAut(G) then L(G)F need not be contractible.
2. Preliminaries 0 If G = G1 G2 Gn, set = G1,...,Gn and = ,G1,...,Gn . For ∗ ∗ · · · ∗ J { 0 } J {∗ } each i, choose an 1 = λn Gi. Let ( ) be the Whitehead poset constructed in [10]. Elements of 6 ( 0)∈ correspondP toJ labelled bipartite trees, where the n + 1 P J labels come from the set ,G1,...,Gn . Often, as in [10], we will abuse notation and take the labels from{∗ the set , 1,}2, . . . , n . Given a labelled tree T and a labelled vertex k in the tree, two other{∗ labels are} equivalent if they are in the same connected component of T k . This gives us a partition A(k) of , 1, 2, . . . , n . The singleton set Q(A(k))− = { }k is called the operative factor of{∗ the partition.} Denote by A the collection of all{ of} these partitions as k ranges over , 1, 2, . . . , n . {∗ } This yields an equivalent notion of elements of ( 0). The poset structure in ( 0) comes from an operation called folding (whenP J the elements are thought of asP J labelled trees) or by setting A(k) B(k) if elements of A(k) are unions of elements of B(k). See McCullough and≤ Miller [10] for more details. Form a deformation retract of ( 0) by folding all edges coming in to on a labelled tree together, resulting in aP labelledJ tree where is a valence 1 vertex.∗ Call the resulting poset P ( ). Observe that elements of P∗( ) correspond to pointed trees with labels in ,J and that P ( ) is (n 1)-dimensional.J Denote elements of P ( ) as pairs ( ,AJ) as in [10]. J − JNow mimic theJ construction in section 2 of [10]. That is, we must construct a space out of the posets P ( ). Let be the set of all bases of G. Define a relation on J B
( ,A) ,A P ( ) { H |H ∈ B ∈ H } by relating ( ,A) and ( ,B) whenever there is a product ρ of symmetric White- head automorphismsH carriedG by ( ,A) so that ρ = and ρA = A. Denote the equivalence class by [ ,A] AnH automorphismH ( , xG) is carried by ( ,A) if = Q(A) = Q(x), x is constantH on each petal of A,H and x is the identityH on the ∗petal 6 containing . The set of all such equivalence classes forms a poset under the folding operation.∗ Denote the poset L(G). Recall that if all of the factors in G are finite, the Kurosh subgroup theorem implies that Aut(G) = ΣAut(G). Define an action of ΣAut(G) on L(G) by having φ ΣAut(G) act via φ [ ,A] = [φ( ), φ(A))]. ∈Recall that the symmetric· H Fouxe-RabinovitchH subgroup ΣFR(G) is the subgroup generated by all symmetric Whitehead automorphisms which do not conjugate their operative factor and that
ΣAut1(G) = ΣFR(G) o Φ, ΣAut(G) = ΣAut1(G) o Ω where Φ = Aut(Gi) is the subgroup of factor automorphisms and Ω (a product of symmetricQ groups) permutes the factors. Further note that Φ and Ω are not 3 canonical. Throughout this entire paper, We make the convention of choosing them to be with respect to the basis 0 = G1,...,Gn . H { }
3. Reductivity lemmas of McCullough and Miller In this section, we sketch how the work of McCullough and Miller in [10] implies that L(G) is contractible. They show K(G) is contractible by defining a norm on nuclear vertices of K(G) and inductively adding the stars of nuclear vertices using this norm while insuring that each new intersection in contractible. We adopt an analogous approach. First, we show that L(G) is contractible using a norm which is directly analogous to that of [10]. In a later section, we will modify this norm along the lines of Krsti´cand Vogtmann in [9] and show that L(G) can also be shown to be contractible with the modified norm. If is a set of elements of G, we can define a norm on nuclear vertices [ , 0] of W H L(G) by setting = w w , where w is the (non-cyclic) word length of w in the basiskHk. W P ∈W | |H | |H To avoid re-doingH work that McCullough and Miller have already done, we adopt the following conventions. Let G be a free product of n finite groups, as already noted. Let G¯ = G λn+1 , where λn+1 = Z/2. There is an injective map from ∗ h i h i ∼ ¯ ¯ ΣAut(G) to ΣOut(G¯) by sending φ ΣAut(G) to φ ΣOut(G¯) where φ(λn+1) = ∈ ∈ λn+1. Moreover, if v = [ ,A] is a vertex in L(G), we can construct a corresponding H vertexv ¯ in K(G¯) by adding λn+1 to and relabelling the vertex in the tree H ∗ corresponding to A as λn+1 (or just n + 1.) Note that if ( , x) is carried by h i H [ ,A] then ( ¯, x¯) is carried by ( ¯, A¯). Finally, if is a set of words in G, H H H W we can construct ¯ by sending w tow ¯ = wλn+1 ¯ (cf. Proposition 2.18 in [5], which isW the basic idea of∈ what W we are doing here∈ W in this adjustment.) Then w v + 1 = w¯ v¯ for any w so that red (α, v) = red ¯ (¯α, v¯) for any α ΣAut| | (G). | | ∈ W W W ∈ Theorem 3.1. The space L(G) is contractible.
Proof. We sketch how the work in Chapters 3 and 4 of McCullough and Miller still applies. Extension of Lemma 3.1 is trivial. For Lemma 3.2, observe that we can define refinements and disjunctions for partitions of 0 as before, and that property 4 on page 27 of [10] implies that if either Q(A) =J or Q(S) = , so that the relevant partition is trivial, then the refinement or disjunction∗ of A ∗with S is just A. Hence analogs of Lemma 3.2 and Lemma 3.3 follow in the context of L(G). Lemma 3.6 and 3.7 of [10] can be used to prove the analogous results in our new context. (The reductive automorphism contructed in Lemma 3.7 still sends λn+1 to λn+1 because of the notions of constricted peak reduction in [5].) For Lemma 3.8 (which McCullough-Miller use to prove their Lemma 4.8 and at the end of their Lemma 4.9) we have symmetric Whitehead automorphisms α, σ at v in L(G) and construct α0, σ0 as in [10]. The complication is that α0, σ0 might not be in L(G) because the petal containing is not conjugated by the identity. We resolve this by conjugating the whole automorphism,∗ if necessary, so that the petal containing does correspond to the identity. More specifically, construct the ∗ correspondingα, ¯ σ,¯ α¯0, σ¯0 in K(G¯). By McCullough-Miller’s Lemma 3.8, we have
red ¯ (¯α0, v) + red ¯ (¯σ0, v) red ¯ (¯α0, v) + red ¯ (¯σ0, v). W W ≥ W W 4
Also, red ¯ (¯α0, v) + red ¯ (¯σ0, v) = red (α0, v) + red (σ0, v) W W W ¯ W by our earlier observations. Now conjugate (in Aut(G))α ¯0, σ¯0 to obtainα ¯0, σ¯0 which are the identity on the petal containing λn+1. Since these only differ by conjugation,
red ¯ (¯α0, v) + red ¯ (¯σ0, v) = red ¯ (¯α0, v) + red ¯ (¯σ0, v) W W W W and we can take the corresponding α0, σ0 in L(G) so that
red (α0, v) + red (σ0, v) red (α0, v) + red (σ0, v), W W ≥ W W proving the analog of the lemma. For Chapter 4, reason as follows. Set 0 = λ1, . . . , λn so that the analog of Lemma 4.1 is that there is only one nuclearW vertex{ of minimal} height n in L(G). Observe that every nuclear vertex in K(G¯) corresponding to a basis of the form i1 i1 in in λ λ1λ− , . . . , λ λnλ− , λn+1 is of minimal height 2n under the basis ¯ 0 = { n+1 n+1 n+1 n+1 } W λ1λn+1, . . . , λnλn+1 . { Lemmas 4.2 and 4.3} are general lemmas about posets from Quillen [12] and hold without any modification. Theorem 4.4 and Proposition 4.5 are the central theorems, established in section 4.2 by the lemmas 4.6, 4.7, 4.8, and 4.9. Lemmas 4.6 and 4.7 can be proven using the same proof. Lemma 4.8 can also be proven using the same proof, even though it uses Lemma 3.8 which has been modified slightly. The same holds for the crucial lemma, Lemma 4.9. The basic idea is that we could think of many of the calculations as taking place in K(G¯), but just with Whitehead automorphisms whose domain (cf. [5] for notions of domain and constricted peak reduction) does not include Gn+1. When we combine and modify these automorphisms, we still obtain ones that are the identity on the last factor. 4. Finite subgroups Proposition 4.1. The stabilizer of a simplex in L(G) under the action of ΣAut(G) is finite. Proof. Let [ ,A] be a vertex of L(G). If i = j, and both ( , xi) and ( , xj) are H 6 H i H j symmetric Whitehead automorphisms carried by [ ,A], then one of xj or xi must be the identity because at least one of the petal of HA(i) containing j or the petal of A(j) containing i also contains . Hence Lemma 7.4 of [10] applies to give us that ∗ ( , xi) and ( , xj) commute. Thus the stabilizer of [ ,A] must be finite. H H H Proposition 4.2. Every finite subgroup of ΣAut1(G) fixes a point of L(G). Proof. ¿From Bridson and Miller in [3], every finite subgroup of Aut(G) fixes a point v0 = [ 0,A0] of K(G). ¿From Theorem 7.6 of [10], any finite subgroup F H of ΣAut1(G) that fixes v0 is conjugate by an inner automorphism µ to a subgroup whose elements are of the form i ( , x )φi Y H i where each φi Aut(Gi),the symmetric Whitehead automorphisms ( , x ) are ∈ i Hi carried by v0, and each xi = 1. Moreover, there is a factor k such that xk = 1 for all i and there is an unlabelled vertex r of the tree T 0 corresponding to A0 such that the petal containing r is always conjugated by the identity in the above Whitehead 5
1 automorphisms. Let = µ− ( 0). Form a tree T by attaching a free edge with H H terminal vertex to T 0 at the vertex r, and let A be the vertex type determined by T . Then F fixes∗ the vertex v = [ ,A] of K(G) and every element of F can be written in the form H i ( , x )φi Y H i where each φi Aut(Gi), the symmetric Whitehead automorphisms ( , x ) are ∈ i i H carried by v, each xi = 1, and there is a factor k such that xk = 1 for all i.
5. Better norm G Well-order G as g1, g2, g3,... and order Z lexicographically. For a nuclear vertex G v corresponding to a basis , define a norm v Z by setting v i to be the H k k ∈ k k (non-cyclic) length gi of gi in the basis . This is analogous to the norm used by Krsti´cand Vogtmann| |H in [9] or Jensen inH [7]. Proposition 5.1. The norm ZG well orders the nuclear vertices of L(G). k · k ∈ Proof. Let U be a nonempty subset of nuclear vertices of L(G) and proceed as in [9]. That is, inductively define Ui and di by setting di to be the minimal length gi obtained by all vertices [ , 0] Ui 1 and letting Ui be all vertices of Ui 1 | |H H ∈ − − which obtain this minimal length. Recall that we chose specific λi Gi. Let N be ∈ such that λi g1, g2, . . . , gN for all i. Let 0 = λ1, . . . , λn . We claim that UN ∈ { } W { } is finite. Let D = g =λ dik so that v 0 D for all v UN . Since each Gi is P ij i k kW ≤ ∈ finite, L(G) is locally finite. Hence the analog of the Existence Lemma 3.7 of [10] implies the ball of radius D (using the 0 distance) around 0 in L(G) is finite. k·kW H So UN is finite. Now choose M N large enough so that g1, g2, . . . , gM contains a representative from each basis≥ element of each basis corresponding{ to an} element of UN . Then UM contains exactly one element, the least element of U.
Let F be a finite subgroup of ΣAut1(G). Our goal in the next few sections is to show that the fixed point subspace L(G)F is contractible. A vertex [ ,A] of L(G)F H is reduced if no element of L(G)F lies below it in the poset ordering. We will show L(G)F is contractible by inductively adding stars of reduced vertices and insuring that intersections are always contractible. The essential step will use the fact that L(G) can be shown the be contractible using the above norm, and we will need the flexibility of being able to well-order G in many different ways. Theorem 5.2. Given any well order of G, the norm v ZG defined above on nuclear vertices of L(G) is such that || || ∈
st(v) ( u Now apply the Existence Lemma 3.7 of [10] with the set of words defined to be g1, g2, . . . , gm, gm+1 to get the desired result. { For Lemma 3.8 (The} Collins-Zieschang Lemma), note that the result is proven in [10] by showing that the inequality holds coordinate-wise in our norm. The arguments given in chaper 4 of [10] also carry through, except that they are simplified somewhat because reductive edges now must be strictly reductive. 6. Fixed point subspaces. If [ ,A] is a vertex type, we think of A as a collection of partitions of , 1, 2, . . . , n H 1 {∗ } rather than a collection of partitions of , where wGiw− in is identified with H H i. For each k, let Ik(A) be the set of labelled vertices that are a distance 2k away from in the tree T corresponding to A. Let I(A) = kIk(A) and define a poset order∗ in I = I(A) by setting r s if r occurs on the minimal∪ path in T from s to ≤ . For i Ik = Ik(A), define I(i) and J(i) as follows. Let z0 = , z1, . . . , zk = i ∗denote the∈ labelled vertices in the unique minimal path from to∗i in T and set ∗ J(i) = (z1, z2, . . . , zk). Let a be the unique unlabelled vertex between zk 1 and − zk and let I(i) = I(a) denote the set of all labels in 1, 2, . . . , n at a distance 1 { } from a in T . That is, I(i) = zk 1 z Ik : zk 1 < z . (Exception: if k = 1, { − } ∪ { ∈ − } let I(i) = z Ik : zk 1 = < z .) Define J j where w = w(J(i)(B)). Now φ = ( , x ) sends λi to Qj H 1 1 1 φ(w− )cwψi(λi)w− c− φ(w) j where c j J j Lemma 6.2. Let [ 0,A] be a vertex type and let φ = j( 0, y )ψj, where each j H j Q H ( 0, y ) is symmetric Whitehead automorphisms, yj = 1 for all j, and each ψj is H F a factor automorphism of Gj. Further suppose that [ 0,A] is reduced in L(G) , H where F = φ . Fix an index k and let d = yj (written so that if j < j j J Proof. Let i be the next labelled vertex on a path from k to in the tree T ∗ corresponding to A. If w j I(k)Gj, ψj(w) = w for all j I(k) i , and i i 1 ∈ ∗ ∈ ∈ − { } ψi(gi) = ykgi(yk)− for all gi Gi occurring in the normal form of w, it is clear 1 ∈ 1 that φ(w) = dwd− . For the other direction, assume φ(w) = dwgkd− and suppose by way of contradiction that w j I(k)Gj. Let S be the set of all indices j for 6∈ ∗ ∈ which an element of Gj is a substring of w. Case 1: There is an index r S I(k) such that J (2) We can get from ( ,C) to ( ,B) by a series of moves conjugating petals K H 1 1 S of various C(i) by w(J(i))πi(w− )w(J(i))− where k S (where the k ∈ w(J(i)) and wk are taken with respect to the ( ,C) representative of v.) K j Proof. For sufficiency, we note that it is a direct check to see that ( 0, y )ψj v = v j H · for all j if v is as described above. So φ = ( 0, y )ψj fixes v as well. Qj H For necessity, suppose that [ ,B] is a reduced vertex in L(G)F . Then φ must fix this vertex type, which meansH that [ ,B] is stabilized by a product H j ( , x )φj Y H j j which equals φ, where xj = 1 for all j and each φj is a factor automorphism of Hj. By (3) of Lemma 6.1, A = B as partitions of , 1, 2, . . . , n . {∗ } We show that the wi have the desired properties by inducting on the distance from i to in T . If i I1, write wi =w ¯ig¯i, wherew ¯i does not end in an element ∗ ∈1 1 1 of Gi. then φ(w ¯iλiw¯i− ) = φ(w ¯i)ψi(λi)φ(w ¯i)− = φi(w ¯iλiw¯i− ). Thus there exists a gi Gi such that φi(w ¯i) = (w ¯i)gi. By Lemma 6.2 gi = 1,w ¯i j I(i)Gj, and ∈ ∈ ∗ ∈ ψj(w ¯i) =w ¯i for all j. By way of contradiction, supposeg ¯i = 1 but ψi(¯gi) =g ¯i. 6 6 Writew ¯i = u1 . . . us in normal form in the basis , where each uj comes from an 1H Hij with ij I1. Because wi =w ¯ig¯i = (w ¯ig¯iw¯i− )u1 . . . us, has length less than s, ∈ 1 1 we have u1 =w ¯ig¯i− w¯i− . Then 1 1 1 1 g¯i− wi− u2 . . . us = ψi(¯gi− )wi− φ(u2 . . . us) Thus if f = φj then Qj 1 1 φ(u2 . . . us) = f(u2 . . . us) = (wiψi(¯gi)¯gi− wi− )u2 . . . us which contradicts the fact that the length of u2 . . . us in is s 1. For the inductive step, consider an index k and let i be theH next− labelled vertex on a path from k to in T . Let w = w(J(i)). As in the proof part (1) of Lemma 6.1, we ∗ j have c j J Applying Lemma 6.2, gk = 1 andw ¯k j I(k)Gj,k◦ . For reasons similar to the base ∈ ∗ ∈ case of the induction, ψk(¯gk) =g ¯k as well. 10 j Corollary 6.4. Let [ 0,A] be a vertex type and let φ = j( 0, y )ψj, where each j H j Q H ( 0, y ) is a symmetric Whitehead automorphism, yj = 1 for all j, and each ψj is H F1 a factor automorphism of Gj. Further suppose that [ 0,A] is reduced in L(G) , H j where F1 = φ . Let F2 be the subgroup generated by all of the ( 0, y )ψj, so that h i FH1 F1 F2. Then some other vertex type [ ,B] is reduced in L(G) if and only if it ⊆ H is reduced in L(G)F2 . F2 F1 Proof. Since F1 F2, L(G) L(G) . However, if v is a nuclear vertex of F1 ⊆ ⊂ L(G) , then from Proposition 6.3 every element of F2 fixes it (i.e, Gj,k◦ = g j j 1 j { ∈ Gj : y g(y )− = ψj(g) only depends on ( 0, y )ψj.) k k } H Let v0 = [ 0,A] be a vertex type, F be a finite subgroup of ΣAut1(G) that fixes H F j v0, and suppose v0 is reduced in L(G) . Let φ = j( 0, y )ψj F , where each j Qj H ∈ ( 0, y ) is symmetric Whitehead automorphisms, yj = 1 for all j, and each ψj is a H j factor automorphism of Gj. Define πj(φ) = ( 0, y )ψj. Define the groups G◦ by H j,k j j 1 j Gj,k◦ = φ F g Gj : ykg(yk)− = ψj(g) where πj(φ) = ( 0, y )ψj . ∩ ∈ { ∈ H } A representative ( ,B) of some other vertex v = [ ,B] is F -standard if all of the following hold H H A = B as partitions of , 1, 2, . . . , n . • {∗ } 1 When we write Hi = w(J(i))Giw(J(i))− then wi j I(i)Gj,k◦ . • ∈ ∗ ∈ Theorem 6.5. Let [ 0,A] be a vertex type and suppose that F ΣAut1(G) fixes H F ⊆ [ 0,A]. Further suppose that [ 0,A] is reduced in L(G) . A necessary and suffi- cientH condition for any other vertexH type v to be reduced is that it have an F -standard representative. Proof. Let F+ be the subgroup generated by πj(φ): j I, φ F so that F F+. { ∈ ∈ } ⊆ Now L(G)F+ = L(G)F by Corollary 6. ¿From Proposition 6.3, each reduced vertex v = [ ,B] in L(G)F+ must have A = B because the structure of B depends only H j on the symmetric Whitehead moves ( 0, y ) occuring in φ F . In particular, B(j) H ∈ j is the wedge (see [10] page 29) of all of the full carriers of the ( 0, y ) occuring in φ F . H ∈ To show that each wi j I(i)Gj,k◦ , note that (2) of Proposition 6.3 means that ∈ ∗ ∈ if the letters from Gj in a particular wk are already in j j 1 j φ F1 g Gj : ykg(yk)− = ψj(g) where πj(φ) = ( 0, y )ψj , ∩ ∈ { ∈ H } then if we take a ξ F and conjugate petals again to get the letters in 6∈ j j 1 j g Gj : y g(y )− = ψj(g) where πj(ξ) = ( 0, y )ψj , { ∈ k k H } then they are still in the previous group and hence in the intersection of the two groups. 7. Contractibility of fixed point subspaces. For this entire section, let F be a finite subgroup of ΣAut1(G) that fixes [ 0,A] F H and suppose that [ 0,A] is reduced in L(G) . If [ ,B] is any other reduced vertex H H 1 and = H1,...,Hn then for all j, k define H◦ = w(J(i))G◦ w(J(i))− . If a is H { } j,k j,k the next unlabelled vertex on a path from k to , set G◦ = G◦ and H◦ = H◦ . ∗ j,a j,k j,a j,k 11 In addition, define Gj,k◦◦ to be Gj,k◦ if it is nontrivial and Z/2 otherwise. Define Hj,k◦◦ , etc., analogously. For each j, a choose 1 = λj,a Gj,a◦◦ . Let Ga = j I(a)Gj,a◦◦ . F 6 ∈ ∗ ∈ Note that if [ ,A] L(G) with ( ,A) F -standard and j I1, then k I1 Hk = H ∈ H ∈ ∗ ∈ k I1 Gk. This follows by letting N be the normal closure of k I1 Gk and considering ∗ ∈ ∗ 6∈ G/N = k I1 Gk. Observe that k I1 Hk N. If j I1, then Hj k I1 Gk because ∼ ∗ ∈ 1 ∗ 6∈ ⊆ ∈ ⊆ ∗ ∈ Hj = wjGjwj− with wj k I1 Gk. Therefore, k I1 Hk k I1 Gk. Furthermore, ∈ ∗ ∈ ∗ ∈ ⊆ ∗ ∈ since is a basis of G, if j I1 and gk Gk, then we can write gk = v1v2 . . . vs in H ∈ ∈ the basis . Taking the quotient by N, this yields a way of writing gk in k I1 Hk. H ∗ 6∈ It follows that k I1 Hk = k I1 Gk, as desired. More generally, one can verify that ∗ ∈ ∗ ∈ 1 k I(a)Hk = k I(a)w(J(i))Gkw(J(i))− . A direct induction argument now yields ∗ ∈ ∗ ∈ that w(J(i)) k I(a) J a AGa ( ,A) Z∪ ∈ k H k ∈ on the F -standard pair ( ,A) representing a nuclear vertex of L(G)F by setting the gth coordinate to be Hg , the length of the word g in the basis given by . As stated, this does not define| |H a norm on nuclear vertices of L(G)F because thereH is more than one way to write the vertex type v = [ ,A] in an F -standard basis. We solve this problem by definining H v = min[ ,A]=v,( ,A) F -standard ( ,A) . k k K K k H k Observe that given any representative ( ,A) there is an easy algorithm to construct H the minimal representative by proceeding inductively through A. If a0 is the least element of A (the vertex adjacent to in T ), then we cannot change the values in ∗ the range Ga0 at all. Supposing we have minimized all values less than a particular a A, we let i be the stem of a. We now conjugate all of I(a) i by a single ∈ 1 − { } element of w(J(i))Gi,a◦ w(J(i))− , if necessary, to reduce the norm restricted to Ga. Proposition 7.2. The norm a AGa v Z∪ ∈ k k ∈ well orders the nuclear vertices of L(G)F . Proof. Let U be a nonempty subset of nuclear vertices of L(G). Inductively define Ug and dg by setting dg to be the minimal length g obtained by all vertices | |H [ ,A] h a k in L(Ga), then let α = ( , x ) be defined as follows: If j i in I or j is not H k ≤ comparable with i in I, then define xj = 1. On the other hand, if j l for some k k 1 ≥ l I(a) i , then set xj = w(J(i))yl w(J(i))− . Suppose α is carried by [ a,B] ∈ − { } a H and Ta is the tree for B. Define T by first cutting out a and all edges attached to a from T , and then glueing Ta in to the resulting hole, by attaching the vertex j of a Ta to the vertex j of T a . Let [ ,B ] be the vertex type corresponding to the tree T a and observe that− {αa}is carriedH by [ ,Ba]. H k Lemma 7.3. If α = ( a, y ) is a reductive Whitehead move at va = [ a, 0] in H a k H L(Ga) carried by [ a,B] as described above, then α = ( , x ) is a reductive Whitehead move atHv = [ ,A] and is carried by [ ,Ba]. H H H Proof. Recall that i is the stem of a. Let w(J(i)) = w. Note that we can assume k y = 1 since λi,a v = 1 is minimal. By way of contradiction, suppose the index k i | | a is one for which Gk,a◦ = 1 . Recall that by Theorem 6.5, wj r I(a)Gr,a◦ for all h i ∈ ∗ ∈ j I(a) i . So when we write each wj in the basis a, we need not use any letters ∈ −{ } 1 H from w(J(i))− Hk,a◦◦ w(J(i)). In addition, if j = k, when we write λj,a in the basis 6 1 a, we need not use any letter from w(J(i))− Hk,a◦◦ w(J(i)). If j = k, the normal H 1 form of λj,a in the basis a uses exactly one letter from w(J(i))− H◦◦ w(J(i)) and H k,a the rest from other elements of a. H Let g = u1,r1 . . . ut,rt be the normal form in the basis va of some element of G , where u G . Then the normal form of g in the basis α(v ) is j I(a) j,a◦ 1,rj r◦j ,a a the∗ ∈ product ∈ [(yk ) 1][(yk u (yk ) 1][yk (yk ) 1][yk u (yk ) 1] r1 − r1 1,r1 r1 − r1 r2 − r2 2,r2 r2 − ... [yk (yk ) 1][yk u (yk ) 1][yk ] rt 1 rt − rt t,rt rt − rt − which has length greater than or equal to t. Furthermore, the length is equal to t only when yk = 1 for every j = 1, 2, . . . t. rj Taking g to be λj,a for k = j I(a), we see that α cannot reduce any of the 6 ∈ lengths λj,a Taking g to be wk, we also see that α cannot reduce λk,a . But α is | | | | reductive by hypothesis and the first coordinates of Ga are the λj,a for j I(a). By k | | ∈ the above paragraph, each yj = 1 and α is the identity map. This is a contradiction. So Gk,a◦ is nontrivial. a k Since α was defined by letting xj = 1 if j i in I or j is not comparable with a ≤ i in I, we know that α does not change any coordinate g with g Gb for some a = b X where (i) b is on the path from i the in T or| (ii)|H where a∈is not on the 6 ∈ ∗ path from b to . Let h Ga give the first coordinate where α is reductive. By a ∗ ∈ Lemma 7.1, α is not reductive on any coordinate of b X Gb up to h, and it is as reductive as α is on the h coordinate. ∪ ∈ k For a particular l I(a) i , xj is constant on the branch of T given by taking j l. Hence αa is carried∈ by− { [ },Ba] in the ascending star of v = [ ,A]. ≥ H H k Note: Observe that there are some cases where a non-reductive move α = ( a, y ) a H at va still induces a well defined (but non-reductive) move α at v. In particular, k we must have Gk,a◦ nontrivial and yi = 1. Next we investigate how a reductive move at v defines moves at various vas. Let α = ( , yk) be a reductive move at v = [ ,A] carried by [ ,B] and let a be an unlabelledH vertex of T which is adjacent toHk. Suppose that YH is the tree for B We k can define the move αa = ( a, x ) at va = [ a, 0] carried by [ ,Ba] as follows: H H H 14 Case 1: a is the next vertex on a path from k to in T . Let i be the stem of k 1 k ∗ a. Set x = w(J(i))− y w(J(i))). The tree Ta for Ba is given by looking at the subtree of Y spanned by vertices in I(a). k 1 k Case 2: k is the stem of a. Set x = w(J(i))− y w(J(i))). As in the previous case, the tree Ta for Ba is given by looking at the subtree of Y spanned by vertices in I(a). Lemma 7.4. If α = ( , yk) is a reductive Whitehead move at v = [ ,A] in L(G)F carried by [ ,B] as describedH above, then there is some a adjacentH to k in the tree H k T determined by A such that αa = ( a, x ) is a reductive Whitehead move at H va = [ a, 0] in L(Ga) carried by [ a,Ba]. H H Moreover, if a0, a1, . . . , am is a complete list of vertices adjacent to k in T then a0 a1 am α = (αa0 ) (αa1 ) ... (αam ) and all of the (not necessarily reductive) terms in the product commute. Proof. The last assertion of the lemma follows directly. It remains to show that at least one αaj is reductive. Let a0 be the vertex adjacent to k in T which is on the path from k to . Let a1, . . . , am be the other vertices adjacent to T , ordered so ∗ k that if ar < as in X then r < s as integers. Assume as always that yk = 1. Let t be k the least index such that yj = 1 for some j I(at). We show that αat is reductive. k 6 ∈ Since y = 1 for some j I(at), α must change the norm of some letter in Ga . j 6 ∈ t By the minimality of t, α does not change the norm of any letter in Ga for a < at. Therefore, since α is reductive, it must be reductive on Gat . Lemma 7.1 now yields that αat is reductive. Theorem 7.5. Let F be a finite subgroup of ΣFR(G) that fixes [ 0,A], suppose F F H that [ 0,A] is reduced in L(G) . Then L(G) is contractible. H Proof. We do this by induction, adding the ascending stars of nuclear vertices v = [ ,A] in L(G)F step by step according to the norm of Proposition 7.2, always H insuring that the reductive part of the star st(v) (of v in L(G)F ) is contractible. We follow the discussion of McCullough and Miller on pages 36-37 of [10]. Namely, we first let R1 be the reductive part of the star of v. We then let R2 denote the full subcomplex of R1 spanned by all vertices each of whose nontrivial based partitions B(i) is reductive. (Note that in the context of vertices in the star of v = [ ,A] in L(G)F , a trivial based partition B(i) is one which is equal to A(i).) H Define a poset map f1 : R1 R2 as follows: If [ ,B] is in R1, then send it to [ ,C], where each nontrivial based→ partition B(j) withH negative reductivity is replacedH by the trivial based partition with the same operative factor. Since f1(w) w for all ≤ w R1, Quillen’s Poset Lemma [12] yields that R2 is a deformation retract of R1. ∈ Let B(k) denote a partition corresponding to some [ ,B] in R2 Suppose that H a0, . . . , am are the vertices adjacent to k in the tree T corresponding to A (cf. the proof of Lemma 7.4.) Then B(k) is admissible if each (B )aj (k) is either trivial aj (that is, equal to A(k)) or reductive. Now let R3 denote the full subcomplex of R2 spanned by all vertices each of whose nontrivial reductive based partitions is admissible. Define a map f2 : R2 R3 by combining all petals of B(k) containing → aj elements of I(aj) k for each j where (B ) (k) is nontrivial and not reductive. − { } aj As before, f2(w) w for all w R2 so that R2 deformation retracts to R3. ≤ ∈ 15 For each a A, let R2(st(va)) denote the reductive portion of the star of va ∈ in L(Ga) where where each based partition is either trivial or reductive. Each nonempty R2(st(va)) is contractible by Theorem 5.2. Let A¯ = a A : R2(st(va)) = { ∈ 6 . Recall that if P1 and P2 are posets, we can form their join P1 ?P2 as the poset ∅} with elements P1 (P1 P2) P2. If p1, p0 P1, p1 < p0 in P1 p2, p0 P2, and ∪ × ∪ 1 ∈ 1 2 ∈ p2 < p0 in P2, then in the poset P1 ?P2 we have (p1, p2) (p0 , p0 ), (p1, p2) p1, 2 ≥ 1 2 ≥ (p1, p2) p2, p1 p10 , and p2 p20 . This coincides with the more usual definition of the join≥ of two≥ topological spaces≥ X [0, 1] Y X?Y = × × (x, 0, y) (x, 0, y ), (x , 1, y) (x, 1, y) ∼ 0 0 ∼ in the sense that the realization of P1 ?P2 is homeomorphic to the join of the realization of P1 with that of P2. However, from Lemmas 7.3 and 7.4 there is a poset isomorphism f3 : R3 ?a A¯R2(st(va)) → ∈ given by f(B) = B . Y a a A,B¯ a=Aa ∈ 6 Since each poset in the join is contractible, ?a A¯R2(st(va)) is contractible. ∈ 8. Permutation automorphisms. When we add permutation groups and consider finite subgroups F of ΣAut(G), the fixed point space is no longer necessarily contractible. Let G1 = G2 = G3 = 1 2 1 1 ∼1 ∼1 ∼ G4 = Z/2 and let φ = ( 0, y )( 0, y )η where y3 = λ1, y1 = y2 = y4 = 1, 2 ∼ 2 2 2 H H y4 = λ2, y1 = y2 = y3 = 1, and η corresponds to the permutation (12)(34) of 0. Let T be the unique labelled tree with d(1, ) = d(2, ) = d(1, 3) = d(2, 4) =H 2 and d(3, ) = d(4, ) = 4, where d( , ) denotes∗ distance∗ in the tree. Let A be the ∗ ∗ · · F vertex type corresponding to T . Then [ 0,A] is a reduced vertex in L(G) , where H F = φ . Further note that no other vertices of L(G)F are in the ascending star of h i [ 0,A]. HNow define [ ,B] by setting H = H1,H2,H3,H4 = λ1λ3G1λ3λ1, λ4G2λ4, λ1λ3λ1G3λ1λ3λ1,G4 H { } { } 1 2 and letting A = B as partitions of , 1, 2, 3, 4 . In addition, set φ0 = ( , x )( , x )ν 1 1 {∗1 1 } 2 2 H2 H2 where x3 = λ1λ3λ1λ3λ1, x1 = x2 = x4 = 1, x4 = λ4λ2λ4, x1 = x2 = x3 = 1, and ν corresponds to the permutation (12)(34) of . Certainly, φ0 fixes [ ,B]. H H However, φ0 = φ because φ0(λ1λ3λ1λ3λ1) = λ4λ2λ4, φ0(λ4λ2λ4) = λ1λ3λ1λ3λ1, φ0(λ1λ3λ1λ3λ1λ3λ1) = λ4λ2λ4λ2λ4, and φ0(λ4) = λ1λ3λ1. So [ ,B] is also in F H F L(G) . It is an isolated vertex not equal to [ 0,A], and hence L(G) is not contractible. H References [1] W. Bogley and S. Krsti´c, String groups and other subgroups of Aut(Fn), preprint. [2] N. Brady, J. McCammond, J. Meier and A. Miller, The pure symmetric automorphisms of a free group form a duality group, J. Algebra 246 (2001) 881-896. [3] M. Bridson and A. Miller, unpublished notes. [4] D. Collins and H. Zieschang, Rescuing the Whitehead method for free products, I: peak reduction, Math. Z. 185 (1984) 487-504. 16 [5] N. D. Gilbert, Presentations of the Automorphism Group of a Free Product, Proc. London Math. Soc. 54 (1987) 115-140. [6] M. Guti´errezand S. Krsti´c, Normal forms for the group of basis-conjugating automorphisms of a free group, International Journal of Algebra and Computation 8 (1998) 631-669. [7] C. Jensen, Contractibility of fixed point sets of Auter space, Topology Appl. 119 (2002) 287- 304. [8] P. Kropholler and G. Mislin, Groups acting on finite-dimensional spaces with finite stabilizers, Comment. Math. Helv. 73 (1998) 122-136. [9] S. Krsti´cand K. Vogtmann, Equivariant outer space and automorphisms of free-by-finite groups, Comment. Math. Helvetici 68 (1993) 216-262. [10] D. McCullough and A. Miller, Symmetric Automorphisms of Free Products, Mem. Amer. Math. Soc. 122 (1996), no. 582. [11] G. Mislin, On the classifying space for proper actions, Cohomological methods in homotopy theory, 263-269, Prog. Math. 196, Birkh¨auser,Basel, 2001. [12] D. Quillen, Homotopy properties of the poset of nontrivial p-subgroups of a group, Adv. in Math. 28 (1978) 101-128. Department of Mathematics, RMIT University, GPO Box 2476V, MELBOURNE 3001, Australia E-mail address: [email protected] Department of Mathematics, The Ohio State University, Columbus, OH 43210 E-mail address: [email protected] Department of Mathematics, University of New Orleans, New Orleans, LA 70148 E-mail address: [email protected]