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Journal of Pure and Applied Algebra 200 (2005) 137–147 www.elsevier.com/locate/jpaa
Every group is an outer automorphism group of a finitely generated groupଁ Inna Bumagina,b,∗, Daniel T. Wisea aDepartment of Mathematics, McGill University, Montreal, Que., Canada H3A 2K6 bSchool of Mathematics and Statistics, Carleton University, Ottawa, Ont., Canada K1S 5B6
Received 3 October 2002; received in revised form20 December2004 Communicated by J. Rhodes Available online 25 February 2005
Abstract We show that every countable group Q is isomorphic to Out(N) where N is a finitely generated C( 1 ) subgroup of a countable 6 small-cancellation group G. Furthermore, when Q is finitely presented, we can choose G to be finitely presented and residually finite. © 2005 Elsevier B.V. All rights reserved.
MSC: 20F28; 20F06; 20F67; 20E26; 20E22
1. Introduction
While many groups arise as automorphisms of algebraic objects, it is not true that every group Q arises as Aut(N) for some group N. Indeed, it is well-known that a nontrivial cyclic group of odd order is not isomorphic to Aut(N) for any group N. The situation for Out(N) is markedly different and the question of whether a given group Q can be realized as an outer automorphism group of some group N has attracted some recent attention. In [11] it was shown that any group Q is isomorphic to Out(N) where N is the fundamental group of a graph of groups but the cardinality of the generating set of N is greater than the
ଁ Research supported by NSERC grant. ∗ Corresponding author. School of Mathematics and Statistics, Carleton University, Ottawa, Ont., Canada K1S 5B6. Fax: +1 613 520 3536. E-mail addresses: [email protected] (I. Bumagin), [email protected] (D.T. Wise).
0022-4049/$ - see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jpaa.2004.12.033 138 I. Bumagin, D.T. Wise / Journal of Pure and Applied Algebra 200 (2005) 137–147 cardinality of the generating set of Q. The problem becomes more complicated if N is required to belong to a special class of groups. Any Q is isomorphic to Out(N) where N is simple [2], or where N is torsion-free metabelian with trivial center [4]. Any countable Q is isomorphic to Out(N) where N is a locally finite p-group [3]. Finally, when Q is finite, N can be chosen to be the fundamental group of a closed hyperbolic 3-manifold [8]. The first main result in this paper is Theorem 11, where we show that any countable Q is (N) C( 1 ) isomorphic to Out , where N is a finitely generated subgroup of a finitely generated 6 group G. The second main result is Theorem 15 where we show that if Q is finitely presented, N can be chosen to be a finitely generated subgroup of a finitely presented residually finite C( 1 ) 6 group. We note that in analogy with Kojima’s result, when Q is finite, the group N C( 1 ) produced by our construction is the fundamental group of a finite 6 complex. The main idea is to use a variant of Rips’s construction [12] in order to obtain a short → N → G → Q → C( 1 ) exact sequence 1 1, where G is a 6 group and N is finitely generated. As observed in [17], the natural homomorphism Q → Out(N) is easily shown to be injective in this case. The main difficulty is to choose G so that there are no “surprise” outer automorphisms of N, and so Q → Out(N) is surjective. A criterion for establishing this surjectivity is given in Lemma 9, which is the central underlying technical result in this paper. We close this introduction by discussing the extent to which our results can be generalized. Since there are uncountably many finitely generated groups Q, it is not true that every such Q arises as Out(N) where N is a subgroup of a finitely presented group. Nevertheless we pose the following problemthat would generalize our two mainresults:
Problem 1. Is every countable group Q isomorphic to Out(N) where N is finitely generated and residually finite?
2. Review of small-cancellation theory
A piece in a presentation a1,a2,... | R1,R2,... is a word U such that US1 = US2 ±1 ±1 and both (US1) and (US2) are cyclic permutations of relators. The presentation satisfies C() if for any piece U which is a subword of the cyclic word URj ,wehave|U| < |Rj | where we use |X| to denote the length of the word X. We will also say that the set of words {R1,R2,...} in the free group a1,a2,... |−satisfies C () if the presentation above satisfies C(). We say that G is a C() group if G has a presentation that satisfies the C() condition. We will need the following well-known facts about the metric small cancellation groups whose presentations satisfy a C() condition. The first theoremis the key result in small cancellation theory.
Proposition 2 (Lyndon and Schupp [10, Chapter V,Theorem 4.4]). If W is a freely reduced C( 1 ) word which represents the identity element in a 6 group, then W contains a subword V such that VS is a conjugate of some relator or its inverse, and S is the concatenation of at most three pieces. I. Bumagin, D.T. Wise / Journal of Pure and Applied Algebra 200 (2005) 137–147 139
C( 1 ) A proof of the following theoremfor 8 groups, based on an idea of Lipschutz [9] C( 1 ) was given by Lyndon and Schupp [10, Chapter V, Theorem10.1] .For 6 groups it was proved by Greendlinger [5].
C( 1 ) Proposition 3. Every torsion element in a 6 small-cancellation group is a power of a conjugate of an element W, where W n is a relator.
C( 1 ) C( 1 ) Greendlinger proved for 8 groups [6], and later for 6 groups [7], that in these groups, any two elements that commute necessarily belong to a cyclic subgroup. Truffault [15] and Seymour [13] independently, strengthened the result of Greendlinger and showed C( 1 ) that in 6 groups, centralizers of non-trivial elements are cyclic. From this latter result we observe that
C( 1 ) G. Corollary 4. LetGbea 6 group, and let N be a non-cyclic subgroup of Then the centralizer of N in G is trivial.
Proof. Let N =h1,h2,.... Then CG(N) ⊂ CG(h1) ∩ CG(h2) ∩···Let d ∈ CG(h1) ∩ CG(h2) ∩···hence hi ∈ CG(d) for all i.Ifd = 1, then CG(d) is cyclic so that H is cyclic, a contradiction. Thus, CG(N) ={1}.
2.1. Construction of special small-cancellation words
Definition 5. An x,y word is special if it is a positive word in x,y with no x2 or y3 subwords.
It is easy to provide sets of special words satisfying stringent small cancellation con- ditions. The following lemma provides us with examples of such sets; we will use these examples in our proof.
Lemma 6. Consider the infinite word W = xy(xy2)xy(xy2)2xy(xy2)3 ···
(1) for each infinite sequence {n1,n2,...} of natural numbers, one can find an infinite family W ={w ,w ,...} C( 1 ) |w |n ∞ 1 2 of words which satisfies the 20 condition, so that i i for each i. (2) for each positive integer m and a large enough integer p one can find a family Vm = {v ,v ,...,v } C( 1 ) 1 2 m of words which satisfies the 20 condition, so that given a subset I of {1, 2,...,m}, the lengths of the elements of Vm are as follows: |vi|=p for each i ∈ I and |vi|=p + 1 for each i/∈ I.
Proof. To prove the first assertion, let
2 ki 2 ki +1 2 2ki wi = xy(xy ) xy(xy ) ···xy(xy ) , (1) k { , 2 n } k { k , 2 n } W . where 1 max 60 3 1 and i max 2 i−1 3 i be the words in ∞ Obviously, w x,y |w |= 9 k2 + k + >n , i is a special word of length i 2 i 5 i 1 i and the length of the maximal 140 I. Bumagin, D.T. Wise / Journal of Pure and Applied Algebra 200 (2005) 137–147
k − k piece pi = y(xy2)2 i 1xy(xy2)2 i xy of wi equals |pi|=12ki + 2. Hence W∞ satisfies the C( 1 ) 20 condition, and we have proven the first assertion. To prove the second assertion, consider the finite subset
Wm ={w2,w3,...,wm+1}⊂W∞ chosen according to part (1) of this lemma (we set ni = 1 for all i). We modify the words wi ∈ Wm as follows. Let p>|wm+1| be an integer number. A “correcting” exponent for i =2, 3,...,m+1, is the maximal integer ci which does not exceed (p −|wi|)/(2(ki +1)). Define (i) (i) (i) ci 2 ki ci 2 ki +1 ci 2 2ki 2 vi = (xy) (xy ) (xy) (xy ) ···(xy) (xy ) x 1 (xy) 2 (xy ) 3 , (i) (i) ∈{, } (i) |v |=p. where 1 , 3 0 1 and 2 is a nonnegative integer, so that i Note that the (i) (i) (i) length of the correcting term x 1 (xy) 2 (xy2) 3 does not exceed 2(ki + 1). V ={v ,v ,...,v } C( 1 ) We claimthat m 1 2 m satisfies the 20 condition. Indeed, the values of ci are all different, hence the maximal length of a piece pi that is a subword of vi is |p ||(xy2)2ki −1(xy)ci (xy2)2ki | < c + k |v | > 9 k2 + c k + c , bounded by i 2 i 12 i. Since i 2 i 2 i i 2 i we have 9 2 9 |vi| ki + 2ciki + 2ci (ki/ci)ki + ki + 1 > 2 = 4 . |pi| 2ci + 12ki 1 + 6ki/ci |p |/|v | < 1 i> . It can be readily seen that i i 20 for all 1 The claimfollows.
3. Outer automorphism groups and short exact sequences
The plan is to use the natural homomorphism Q → Out(N), which exists for any short exact sequence 1 → N → G → Q → 1. (2)
Specifically, G acts by conjugation on N, so there is a homomorphism ˆ : G → Aut(N), and obviously ˆ (N) ⊂ Inn(N), so there is an induced homomorphism : Q → Aut(N)/Inn(N) = Out(N). Our goal is as follows: for a given group Q, find G and N in (2) so that : Q → Out(N) (3) is an isomorphism.
3.1. The map is injective
We will prove that is injective by applying the following simple criterion which holds for an arbitrary extension (2).
Lemma 7. If the centralizer CG(N) of N in G is contained in N, then is injective. I. Bumagin, D.T. Wise / Journal of Pure and Applied Algebra 200 (2005) 137–147 141
Proof. The kernel of is represented by elements g ∈ G which act on N like an inner automorphism, say conjugation by n ∈ N. Specifically, for each x ∈ N,wehavexn = xg −1 and so xng = x. Thus ng−1 is in the centralizer of N, and so ng−1 ∈ N, and hence g ∈ N as claimed. Thus g represents a trivial element of Q.
In our construction, G will be a non-cyclic small cancellation group and N will be a 2-generated subgroup of G. Therefore, by Corollary 4, the centralizer of N is trivial, so it is obviously contained in N, and we have the following statement.
C( 1 ) Lemma 8. If the group G in (2) is non-cyclic and 6 , then is injective.
3.2. The map is surjective
Given a group Q presented by A | R, where A ={ai}i∈I and R ={Rj }j∈J, we form a presentation a a−1 a a−1 x,y,A | x i = Ai,xi = Bi,yi = Ci,yi = Di,i∈ I,Rj = Ej , p p q j ∈ J,x,y,xyxy2xy3 ···xy , (4) where the Ai,Bi,Ci,Di and Ej are words in x and y that are specified in Lemma 9(2) below. Letting G denote the group presented above, we obtain a short exact sequence 1 → N → G → Q → 1 where N =x,y. The first four types of relations guarantee that N is normal in G, and the Rj =Ej relations guarantee that G/N Q. The last three relations are added to facilitate the arguments determining Out(N) that we shall make below. Of course, a priori, there is no reason to conclude that N is nontrivial. In order to obtain a group G with the desired properties, we use small cancellation techniques.
Lemma 9. If Presentation (4) for group G in (2) satisfies the following conditions, then the map : Q → Out(N) is surjective. C( 1 ) (1) Presentation (4) satisfies the 11 condition. (2) The relations of G are chosen so that Ai,Bi,Ci,Di and Ei are special x,y words, and 30 each such special word has length 31 of its corresponding relator. (3) Exponent q is an integer satisfying q>45. (4) Exponent p is a number satisfying p>2q.
Proof. Pick ∈ Aut(N). Since the only relators in G which are proper powers are xp and yp, by Proposition 3, the only torsion elements in G are conjugates of powers of x or y. p p (x) = g−1z 1 g (y) = g−1z 2 g g ∈ G, z ∈{x,y} Therefore, 1 1 1, and 2 2 2, where each i each i , and 0pi
p p ~ 1 ~ 2 z1 z 2
S
g~
Fig. 1. (N) is the fundamental group of the bold graph. The thin path represents a relator VS.
We shall later prove that ∈/ Aut(N) unless (x) = x and (y) = y. Our arguments show that in all the other cases the subgroup (x), (y) is isomorphic with (x)∗(y) and therefore cannot be isomorphic to N. −1 p1 −1 p2 We argue as follows. We may assume that the words g z g1 and g z g2 are freely q q 1 1 2 2 g = g . g g−1 = z 1 z 2 q ,q , g = reduced. Suppose that 1 2 If 1 2 1 2 for some integers 1 2 then 1 q q − p − −q p q z 1 z 2 g , hence g 1z 1 g = g 1z 2 z 1 z 2 g . We see that is conjugation by g pre- 1 2 2 1 1 1 q2 2 1 2 2 2 composed by conjugation by z 2 and by the map ; in this case, we can proceed to the next 2 q q g g−1 = z 1 z 2 q ,q . step. Therefore, we may assume that 1 2 1 2 for any integers 1 2 Consider the free group F =˜x,y,˜ a˜1, a˜2,... |−and the homomorphism : F → G that sends x˜ to x, y˜ a˜ a i ∈ I. g,˜ z˜ , z˜ ∈ F (g)˜ = g g−1, to y and i to i for each Pick the shortest 1 2 so that 1 2 ˜ −1 (z˜i) = zi, and consider the subgroup K of F generated by z˜1 and g˜z˜2g˜ . We claimthat the image K = (K)˜ in G is a free product of two cyclic subgroups and therefore cannot be isomorphic to N.Indeed, consider the graph (see [14]) illustrated in Fig. 1 corresponding to the generators of K.˜ The graph is represented by the bold part of Fig. 1. It has two p p z˜ 1 z˜ 2 ), g.˜ closed paths (labelled by 1 and 2 which are connected by an arc labelled by In the −1 product g˜z˜2g˜ , absorb the last z˜2 syllable of g˜ if necessary, and change the base point of if g˜ starts with z˜1. By the above observations, we can assume that g˜ starts with a letter ˜ different from z˜1 and ends by a letter different from z˜2. If the intersection K ∩ ker() is not trivial, then by Proposition 2, there is k˜ ∈ K˜ that contains a subword V such that VS is a conjugate of some relator or its inverse, and S is the concatenation of at most three pieces. In particular, this large part V of a relator is represented by a path in the graph. Unless V is either xn or yn for some 0 |˜g||U|−|S|− − 2 |U| > |U|− − 5 |U| > 5 |U| 4 11 4 11 11 ; I. Bumagin, D.T. Wise / Journal of Pure and Applied Algebra 200 (2005) 137–147 143 the last inequality holds for U>44. Hence, g˜ can be replaced by the shorter path which consists of S and two pieces. In the second case, observe that our choice of the relations implies that V cannot travel through the entire arc g˜ more than twice. Indeed, each occurrence of g˜ in V is followed by an occurrence of g˜−1. Since both x and y appear in the relators only with positive exponents, ±1 and g˜ is nontrivial, g˜ necessarily contains a˜i for some i. The structure of relators implies that ± ± ± ∓ ± V =˜g z 2g˜ 1z 2g˜ 1z 2g˜ , 1 i1 i2 i1 2 where g˜1 and g˜2 are subpaths of g˜. Observe that g,˜ g˜1 and g˜2 are pieces. Altogether we |V | < 4 |U|+ < 1 |U|, |V | < 5 |U| have 11 6 and since 6 11 we obtain the following inequality 11 that contradicts Proposition 2. Therefore, each k˜ ∈ K˜ ∩ ker() contains more than half of r r −1 x or y . Let w˜ ∈ F be a word of length at least two in the free product ˜z1∗˜gz˜2g˜ , and let w˜ =˜w1 ···w ˜s be its normal form. Our arguments imply that (w)˜ is non-trivial in G unless (w˜i) = 1G for each i = 1,...,s. The claimfollows. p p (x) = g−1z 1 g (y) = g−1z 2 g . p = We have shown that 1 1 1, and 1 2 1 We now show that 1 p2 =1. We are choosing shortest representatives subject to 0 − p p p p p qp W = g (W)g 1 = x 1 y 2 x 1 y2 2 ···x 1 y 2 1 1 . Since (W)=G1, we have W =G1, and Proposition 2 implies that W contains more than half of a relator. Since we have chosen shortest representatives for (x) and (y), the word W does not contain more than half of the relator xp. The other relators are words with n no x subwords for |n| > 1; therefore, p1 = 1. Note that if we started with a negative value of p1, then we would actually have p + p1 = 1, so that p1 =−(p − 1). But then g−1xg (x) 1 1 should be the shortest representative for . Furthermore, in all cyclic conjugates ±1 ±1 ±1 ±1 ±1 of Ai ,Bi ,Ci ,Di and Ei the generator y appears with exponents ±1 and ±2. Assume that the exponents of x and y in W are positive. For each m with 1