J. Group Theory Galley Proof, 9 pages DOI 10.1515/jgth-2015-0006 © de Gruyter 2015

Bounded generation of wreath products

Nikolay Nikolov and B. Sury Communicated by Miklós Abért

Abstract. We show that the (standard restricted) wreath product of groups is boundedly generated if and only if the bottom group is boundedly generated and the top group is finite.

1 Introduction

Wreath products of groups (for definitions and notation see Section 2) naturally arise in the study of Sylow subgroups of appropriate symmetric groups. They also often provide examples of certain groups with rather unexpected properties. The goal of this paper is to investigate bounded generation of (standard restricted) wreath products of groups. An abstract group G is said to have bounded generation if there exist (not neces- sarily distinct) elements g1; : : : ; g G such that G g1 g , where gi k 2 D h i


h ki h i is the cyclic subgroup generated by gi . Even though defined as a simple combi- natorial notion, bounded generation turns out to imply a number of remarkable structural properties:

 the pro-p completion of a boundedly generated group is a p-adic analytic group for every prime p ([2, 6]);

 if a boundedly generated group G has property (FAb), then it has only finitely many inequivalent completely reducible representations in every dimension over any field (see [7, 13, 14] for representations in characteristic zero, and [1] for arbitrary characteristic);

 if G is a boundedly generated S-arithmetic subgroup of an absolutely simple simply connected algebraic group over a number field, then G has the congru- ence subgroup property ([8, 11]);

 bounded generation can be used to find explicit Kazhdan constants ([3, 15]). More examples of boundedly generated groups are provided by the finitely gen- erated soluble groups of finite rank [5]. It is an open problem whether or not the converse holds: Does every soluble boundedly generated group G have finite rank? 2 N. Nikolov and B. Sury

An affirmative answer has been obtained in [12] in the case when G is residually finite. Further progress on this problem will certainly involve better understanding of modules for soluble minimax groups. While we are unable to contribute to this problem, in this note we establish the following criterion for bounded generation of wreath products.

Theorem 1.1. If A and B are nontrivial groups, then A B has bounded genera- o tion if and only if A has bounded generation and B is finite. We remark that the question of bounded generation of complete wreath prod- ucts is trivial: if B is finite, then the complete wreath product is the same as the restricted wreath product, and if B is infinite, then the complete wreath product A Wr B is not finitely generated, hence cannot be boundedly generated.

2 Preliminaries

Wreath products are defined in the context of permutation groups when a group A acts on a set X and a group B acts on a set Y . We will consider the so-called standard wreath products, in which case the groups A and B act on themselves via right regular representations. Moreover, we will work with restricted standard wreath products, which are defined as follows. We will use some notations and terminology from [9]. Let A.B/ be the direct sum of copies of A indexed by elements of B (for the complete wreath product one takes the direct product). We will represent elements .B/ of the group A as tuples a .ab/b B and will refer to ab A as the coordinate D 2 2 of a at b B. The set 2 .a/ b B a 1 B D ¹ 2 j b ¤ º Â is the support of a; notice that .a/ is finite for every a A.B/, and define the 2 length of a by `.a/ .a/ . D j j The (standard restricted) wreath product of A and B, denoted by A B, is the o semidirect product of A.B/ and B with the action of B on A.B/ given by

b0 a .cb/b B ; where cb abb 1 : (2.1) D 2 D 0 The group A is called the bottom group, the group B is called the top group, and the group A.B/ is called the base group. An element a A.B/ with the property .a/ b will be denoted by a , in 2 D ¹ º b other words, the only nontrivial coordinate of ab is ab. When it is important to emphasize that a particular element ai A is the coordinate of a at b, we will 2 b slightly abuse notation and write Œai b for ab. In this notation, every element a Bounded generation of wreath products 3 of A.B/ can be written in the form a a a a Œa  Œa  Œa  ; D b1 b2


bs D b1 b1 b2 b2


bs bs where b1; b2; : : : ; bs are distinct elements of B; observe that s `.a/. If we iden- D tify elements of B with their copies in A B, then (2.1) yields the following o relation in the wreath product:

b0a Œa  b0; where b0; b B; a A: b D b bb0 2 b 2 We will represent elements of the wreath product A B as products ab, where o a A.B/ is an element of the base group and b B is an element of the top group. 2 2 Next, we will need the following standard facts about bounded generation for the proof of Theorem 1.1.

Lemma 2.1 ([16]). Let G be a group and H be its subgroup. (i) If H has finite index in G, then bounded generation of G is equivalent to that of H. (ii) If G has bounded generation, then so does any homomorphic image of G. (iii) If H is a normal subgroup of G and both H and G=H have bounded gener- ation, then so does G.

Proof. Part (ii) is immediate from the definitions. For part (iii) observe that if H h1 hs and G=H g1H gt H D h i


h i D h i


h i for some hi H and gi G, then G h1 hs g1 gt . Note 1: 2 2 D h i


h ih i


h i Red parts For part (i) in the case when H h1 ha has bounded generation we indicate D h i


h i major choose a set g1; : : : ; gb of coset representatives for H in G and then changes. Please check them G h1 ha g1 g : carefully. D h i


h ih i


h bi So the only part remaining to prove is that bounded generation of G implies that of its subgroup H of finite index. Suppose that G g1 gn . Since every D h i


h i subgroup of finite index contains a normal subgroup of finite index, it suffices to prove our claim assuming in addition that H is normal in G. r1 rn An arbitrary element h H can be written in the form h g gn for 2 D 1


some r1; : : : ; rn Z. Say, ŒG H  m and for i 1; : : : ; n write ri ei mai 2 W D D m D C with 0 6 ei < m. Since H is normal, we have hi g H. In this notation, WD i 2 h gr1 grn ge1 ha1 ge2 ha2 gen han D 1


n D 1 1 2 2


n n n 1 ! e1 en Y ei 1 en 1 ei 1 en ai an g g Œ.g C g / hi .g C g / h : (2.2) D 1


n i 1


n i 1


n n i 1 C C D 4 N. Nikolov and B. Sury

We now introduce the following finite set:

e1 en ƒ g g 0 6 ei < m : D ¹ 1


n j º Then it follows from (2.2) that  à n  à Y Y Y 1 H y x hi x ; D h i
h i y ƒ H i 1 x ƒ 2 \ D 2 so H has bounded generation.

Finally, we will need the following result from [10].

Theorem 2.2. Suppose that a group G is a union of n (left or right) cosets of its subgroups H1;:::;Hk. Then one of the subgroups Hi must have index at most n in G.

Corollary 2.3. Let G be a group and H1;:::;Hk be subgroups of infinite index in G. Then for every integer n there exist n elements a1; : : : ; an in G such that the set a1; : : : ; an cannot be covered by fewer than n (left or right) cosets of ¹ º H1;:::;Hk.

Proof. We will construct elements a1; : : : ; an no two of which are in the same left or right coset of any Hi by induction on n. This is trivial for n 1. Suppose D that we have found elements a1; : : : ; am with the above property. Since the sub- groups H1;:::;Hk have infinite index in G, Theorem 2.2 implies [ G aj Hi ;Hi aj j 1; : : : ; m; i 1; : : : ; k : ¤ ¹ j D D º Now take any element of G outside the right hand side and call it am 1. C

3 Proof of Theorem 1.1

If B is finite, then the base group A.B/ has finite index in the wreath product A B o and thus bounded generation of A B is equivalent to that of A.B/ (Lemma 2.1). o On the other hand, A.B/ is a direct sum of finitely many copies of A, hence it has bounded generation if and only if A does. We have reduced the proof of Theorem 1.1 to the proof of the following statement.

Theorem 3.1. If A is a nontrivial group and B is an infinite group, then the wreath product A B is not boundedly generated. o Assume on the contrary that A B is boundedly generated: A B C1 C o o D


k where Ci ai bi is a cyclic subgroup generated by ai bi A B. Let Bi bi D h i 2 o D h i Bounded generation of wreath products 5

be the cyclic subgroup of B generated by bi , 1 6 i 6 k; observe that

B B1 B : D


k Lemma 3.2. The group B is virtually cyclic.

Proof. Assume the contrary; then all subgroups B1;:::;Bk have infinite index Pk in B. Let N k `.ai /. By Corollary 2.3 there is a finite subset S of B D i 1 which is not coveredD by any collection of N cosets of the following finite set of subgroups of B

´ ˇ k µ 1 ˇ [ B xBi x ˇ x .aj /; i 1; : : : ; k : D ˇ 2 D ˇ j 1 D Let d be an element of A.B/ whose support contains the set S. We have

k Y ni d .ai bi / (3.1) D i 1 D n ni for some integers ni Z, 1 6 i 6 k. Write .ai bi / i ei b , where 2 D i 2 ni 1 ´ b b b .ai / . i ai / . i ai / . i ai / if ni > 0; ei




n b 1 1 b 2 1 b i 1 D . i a / . i a / . i a / if ni < 0: i
i


i It follows that the support .ei / of ei is contained in the union of the left cosets xBi x .aj /; i; j 1; : : : ; k of B1;:::;B . On the other hand, (3.1) gives ¹ j 2 D º k p1 p2 pk 1 n1 nj d .e1/ . e2/ . e3/ . e /; where pj b b ; D




k D 1


j which shows that the support of d is contained in the union of at most N right 1 1 cosets xBi pj .xBi x /xpj of the subgroups xBi x B. This is a contra- D 2 diction. Thus, one of the subgroups Bi has finite index in B and so B is a virtually cyclic group.

Lemma 3.3. It suffices to prove that A Z is not boundedly generated. o Proof. We proved that B contains a cyclic subgroup, say, B0 Z of finite index. .B/ ' Now A B contains the semidirect product A Ì B0 as a subgroup of finite index Note 2: o .B/ Removed and therefore A Ì B0 is boundedly generated by Lemma 2.1 (i). In turn the map ‘the wreath .B/ .B / .B/ .B / product’ A Ì B0 A 0 Ì B0 defined by the canonical projection A A 0 for ! ! the base groups and identity on the top group B0 is easily seen to be surjective .B / homomorphism. Hence A 0 Ì B0 A Z is boundedly generated as a quotient D o of a boundedly generated group by Lemma 2.1 (ii). 6 N. Nikolov and B. Sury

From now on we assume that B is an infinite cyclic group and we will simply index the coordinates of elements a A.B/ by integers; the generator z of B then 2 acts on A.B/ as the shift i i 1 of the coordinates. In this new notation we have 7! C t the assumption that A B C1 C , where Ci ai z i , ti Z, 1 6 i 6 k. o D


k D h i 2 Lemma 3.4. The group A is finite. Proof. We let J be the subset of 1; 2; : : : ; k consisting of those i for which ¹ º ti 0, also let I be the complement of J in 1; 2; : : : ; k ; observe that J ¿. ¤ ti ˛i ¹ ˛i º ¤ If i I , then the power .ai z / , ˛i Z, is simply ai . If i J , then the power 2t ˛ 2 2 .ai z i / i is a product of the following element a of the base group,

´ ti 2ti .˛i 1/ti .a / .z a / .z a / .z a / if ˛ > 0; a i i i i i (3.2) z ti
1 z
2ti 1


z˛i ti 1 D . a / . a / . a / if ˛i < 0; i
i


i ˛ t z˛i and the element z i i of the top group. If ˛i is large enough, then ai and ai z˛i j j have disjoint support since . ai / ˛i .ai /. It follows that each coordinate D C of the elements in (3.2) is a product of boundedly many coordinates of ai . There- fore there is a constant Mi which depends only on ai and ti , but not on ˛i , such that the coordinates of the elements in (3.2) take at most Mi values in A. .B/ Since we assume that A B C1 C , every element of the base group A o D


k is a product of at most k of some B-conjugates of elements as in (3.2) and at most k of some B-conjugates of elements of the form a˛i . Notice that any B-conjugate ˇ i z i ˛i ai has support of size `.ai /, so the total support of B-conjugates of such Pk elements corresponding to i I has size at most i 1 `.ai /. If we take an ele- .B/ 2 Pk D ment a of A of length `.a/ > i 1 `.ai /, then at least one of its coordinates has to be a product of coordinates ofD elements of the form (3.2) corresponding Q to i J . But such a product can take at most i J Mi values in A. It follows 2 Q 2 that A cannot have more than i J Mi elements. 2 Remark 3.5. One can also conclude that A must be finite in the original set-up, without the assumption that B is infinite cyclic. If we similarly partition the set 1; 2; : : : ; k into two subsets I (consisting of those i for which bi has finite order) ¹ º and J (consisting of those i for which bi has infinite order), then the support of all B-conjugates of elements corresponding to I is bounded, while the coordinates of B-conjugates of elements corresponding to J take a bounded number of values in A.

We have now reduced Theorem 3.1 to the following case: A is finite and B is infinite cyclic. Define the width of a A.B/, denoted w.a/, as the difference 2 between the largest and the smallest element of the support of a and let

L max w.a1/; : : : ; w.a / : D ¹ k º Bounded generation of wreath products 7

For a large positive integer M let  A.B/ AM be the projection onto the coor- W ! dinates 1; 2; : : : ; M : ..ai /i Z/ .a1; : : : ; aM /: 2 D Take an arbitrary element azt A B with w.a/ 6 L and t 0. For an inte- 2 o ¤ ger ˇ1, a positive integer ˇ2, and  1 , consider the element 2 ¹˙ º ˇ1 ˇ1 t ˇ1 2t ˇ1 ˇ2t g .z a/ .z C a/ .z C a/ .z C a/: ˇ1;ˇ2; D




zˇ1 it Let ui Z be the position of the first nontrivial coordinate of the element C a, 2 0 6 i 6 ˇ2. More explicitly, assuming  1, ui u0 it and D D C zˇ1 it . C a/ u0 it; u0 1 it; : : : ; u0 L it : Â ¹ C C C C C º

The number of possible projections of gˇ1;ˇ2; as ˇ1, ˇ2,  vary depends only on a, t, and the coordinates whose positions are between 1 and M . More pre- zˇ1 it cisely, since ui determines the element C a, once a and t 0 are fixed, the ¤ projection .gˇ1;ˇ2;/ is completely determined by the smallest and the largest of the positions u0; u1; : : : ; u which lie in the interval Œ .L 1/; M . We have ˇ2 therefore proved the following:

Lemma 3.6. For any given ai and a choice of  1 we have 2 ¹˙ º 2 .g / ˇ1 Z; ˇ2 N .M L/ j¹ ˇ1;ˇ2; j 2 2 ºj Ä C As observed earlier, every element of A.B/ is a product of at most k of some B-conjugates of elements of the following three types:

t .˛ 1/t t 2t ˛ t z i z i i z i 1 z i 1 z i i 1 ˛i .ai / . ai / . ai /; . a / . a / . a /; a : (3.3)



i
i


i i We have just established that there are at most .M L/2 possibilities for the C projections onto AM for each of the elements of the first two types in (3.3).

Lemma 3.7. For any given ai we have .z a˛z / ; ˛ Z .M L/ A `.ai /: j¹ i j 2 ºj Ä C j j ˛ Proof. Observe that the B-conjugates of ai can have only `.ai / nontrivial coor- dinates, which are a shift of the nontrivial coordinates of ai . To count the number of possible projections of such elements, we notice that there are at most A `.ai / j j possibilities for their nontrivial coordinates, and by shifting the position of the first nontrivial coordinate (which can take any value between .L 1/ and M ) we obtain at most .M L/ possibilities for each combination of coordinates. C It follows that the projections of the elements of the third type from (3.3) can take at most .M L/ A `.ai / values in AM . C j j 8 N. Nikolov and B. Sury

Putting the last two lemmas together, we see that the product of k B-conjugates of elements as in (3.3) can have at most ..M L/2/k ..M L/ A D/k C
C j j values in its projection under , where D max `.a1/; : : : ; `.a / . This number D ¹ k º grows polynomially in M when L, D, k, and A are fixed. On the other hand, j j A M grows exponentially with M since A > 1. So for large enough M we will j j j j not be able to achieve all the choices for the coordinates 1; 2; : : : ; M with a product of k cyclic subgroups Ci . This contradiction completes the proof of Theorem 3.1.

Remark 3.8. The proof of Theorem 3.1 can be adapted to some other groups beyond standard wreath products. For example it shows that, when A 1, then On 6D the group A S Ì SLn.OS / is not boundedly generated, where OS is the ring of S-integers in a number field.

Acknowledgments. This work was begun during the second author’s visit to the University of North Carolina at Greensboro and was completed after his visit to the Imperial College London. He would like to thank both these institutions for their excellent hospitality.

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Received May 20, 2012; revised January 12, 2015.

Author information Nikolay Nikolov, Mathematical Institute Oxford, OX2 6GG, UK. E-mail: [email protected] B. Sury, Stat–Math Unit, Indian Statistical Institute, 8th Mile Mysore Rd, Bangalore 560 059, India. E-mail: [email protected]