Generating groups by conjugation-invariant sets
aa r X i v : . [ m a t h . G R ] M a y GENERATING GROUPS BY CONJUGATION-INVARIANTSETS
VALERY BARDAKOV, VLADIMIR TOLSTYKH, VLADIMIR VERSHININ
Abstract.
Let S be a generating set of a group G. We say that G has finitewidth relative to S if G = ( S ∪ S − ) k for a suitable natural number k. We saythat a group G is a group of finite C -width if G has finite width with respect toall conjugation-invariant generating sets of G. We give a number of examplesof groups of finite C -width, and, in particular, we prove that the commutatorsubgroup F ′ of Thompson’s group F is a group of finite C -width. We alsostudy the behaviour of the class of all groups of finite C -width under somegroup-theoretic constructions; it is established, for instance, that this class isclosed under formation of group extensions. Introduction
Let G be a group and let S be a generating set of G. Then the length | g | S ofan element g ∈ G with regard to S is the least number of elements of S ∪ S − whose product is g and the width wid( G, S ) of G with regard to S iswid( G, S ) = sup {| g | S : g ∈ G } . Thus wid(
G, S ) is either the least natural number k such that every element of G is written as a product of at most k elements of S ∪ S − , or wid( G, S ) = ∞ ifsuch k does not exist. If the width G with regard to S is a finite number k, it isconvenient to say that G is generated by S in k steps. If, furthermore, S = S − is a symmetric generating set, then G = S k . Recall that a group G is said to be a group of finite width (or to satisfy the Bergman property ), if G has finite width with respect to every generating set. Thefirst example of an infinite group of finite width has been given in 1980 by S. She-lah [16]. In 2003 Bergman (see [5]) proved that all infinite symmetric groups aregroups of finite width. Some questions Bergman included in his 2003 preprint of[5] initiated the search of new examples of groups of finite width. In particular, ithas been discovered that the following groups are groups of finite width: the au-tomorphism groups of 2-transitive linearly ordered sets [9], ω -existentially closedgroups [8], the autohomeomorphism groups of some topological spaces [10], theautomorphism groups of infinite-dimensional vector spaces over arbitrary skew Mathematics Subject Classification. fields [17], the automorphism groups of infinitely generated free nilpotent groups[18], the automorphism groups of many ω -stable and ω -categorical structures [11]and so on.The class of all groups of finite width has a number of attractive propertiesbeing, for instance, closed under homomorphic images and under formation ofgroup extensions [5].Naturally, one can weaken the condition requiring that a given group G havefinite width with regard to all generating sets (which leads to the notion of agroup of finite width) by requiring that G have finite width with regard to allgenerating sets satisfying a certain condition χ, thereby getting the notion of agroup of finite χ -width.In the present paper we study one of the natural choices for χ above, thecondition C = ‘to be conjugation-invariant’ (to be invariant under all inner auto-morphisms). Thus we say that a given group G is a group of finite C -width if G has finite width with regard to every conjugation-invariant generating set. Everygroup of finite width is of course a group of finite C -width, but the converse isnot true (since, for instance, there are finitely generated infinite groups of finite C -width). The notion of a group of finite width and the notion of a group offinite C -width coincide on the class of all abelian groups; Bergman [5] provedthat the abelian groups of finite width are exactly the finite ones.It is easy to see that the special linear group SL n ( K ) where n > K is an arbitrary field is a group of finite C -width. Indeed, the conjugacy class T ⊆ SL n ( K ) which consists of all transvections generates the group G in n steps:SL n ( F ) = T n . On the other hand, if S = S − is a symmetric, conjugation-invariant generating set of SL n ( K ) , then an appropriate power S k contains atransvection (and hence all transvections), whence SL n ( K ) = ( S k ) n = T n = S kn . In the next section we shall give some other examples of groups of finite C -width. In particular, we shall prove that the commutator subgroup F ′ of Thomp-son’s group F is a group of finite C -width. The proof uses the ideas developedin the paper [6] by Burago, Ivanov and Polterovich. We shall also show thatsome groups of ‘bounded’ automorphisms of infinitely generated relatively freealgebras are groups of finite C -width.In the final section we analyze the behavior of the class of all groups of finite C -width under some group-theoretic constructions. It shall be proved that theclass of all groups of finite C -width is closed, like the class of all groups of finitewidth, under formation of group extensions. We shall consider the question whena free product of nonidentity groups is a group of finite C -width. It turns outthat the only, up to an isomorphism, free product of nonidentity groups which isa group of finite C -width is the group Z ∗ Z . We also give a number of necessaryand sufficient conditions for all functions L : G → R on a given group G such ENERATING GROUPS BY CONJUGATION-INVARIANT SETS 3 that L − (0) = { } ,L ( g − ) = L ( g ) ( g ∈ G ) ,L ( gh ) L ( g ) + L ( h ) ( g, h ∈ G ) ,L ( hgh − ) = L ( g ) ( g, h ∈ G ) , called norms in [6], to be bounded from above. For instance, all norms on a givengroup G are bounded from above if and only if G is a group of finite C -width andevery exhaustive chain ( N k ) of normal subgroups of G terminates after finitelymany steps. One more such a condition states that for every action of G byisometries on a metric space h M ; d i such that d ( ghg − a, a ) = d ( ha, a ) ( a ∈ M ; g, h ∈ G ) , the diameters of all G -orbits are bounded.2. Examples of groups of finite C -width We shall say that a generating set S of a given group G is conjugation-invariant ,if S is invariant under all conjugations (inner automorphisms of G ). A group G is said to be a group of finite C -width if G has finite width relative to everyconjugation-invariant generating set.The following simple sufficient condition for finiteness of C -width works, as weshall see, in many cases. Lemma 2.1.
Let a group G be generated by finitely many conjugacy classes C , . . . , C m in finitely many steps: wid( G, C ∪ . . . ∪ C m ) = N < ∞ . Then G is a group of finite C -width.Proof. Suppose that C k = a Gk is the conjugacy class of a certain element a k ∈ G ( k = 1 , . . . , m ) . Consider a symmetric, conjugation-invariant generating set S of G. For every k = 1 , . . . , m, a suitable power S p k of S contains the element a k . Set p = max( p , . . . , p m ) . Then C ∪ . . . ∪ C m ⊆ S p , and it follows that G = ( S p ) N = S pN . (cid:3) Now let us consider examples of groups of finite C -width. ENERATING GROUPS BY CONJUGATION-INVARIANT SETS 4
1) We have already mentioned in the introduction that the special linear groupSL n ( K ) over a field K of dimension n > C -width, since thisgroup is generated by the conjugacy class of all transvections in n steps.2) Lemma 2.1 and Example 1.1 from [6] imply that the special linear groupSL n ( Z ) where n > Z is also a group of finite C -width.In fact this result has been proven in [6] (though in different terms). Indeed,it is known that the group SL n ( Z ) is generated by the family of all elementarytransvections in some K n < ∞ steps (see [14]). An elementary transvection t ij ( m ) , where i, j with 1 i, j n are distinct indices and m ∈ Z , is the matrix t ij ( m ) = I + mI ij . Observe that the group SL n ( Z ) is also generated by the conjugacy class T of thetransvection t (1) . On the other hand, the following well-known formula t ij ( m ) = [ t ik (1) , t kj ( m )] ( m ∈ Z ) , where 1 i, j, k n are pairwise distinct indices, implies that every elementarytransvection is a product of two elements of T. Thus the group SL n ( Z ) is generatedby the conjugacy class T in at most 2 K n steps.3) Among known examples of groups of finite width, quite a few are groups thatare generated by a single conjugacy class. For instance, the mentioned propertyis shared by the infinite symmetric groups, the automorphism groups of infinitedimensional vector spaces over skew fields, the automorphism groups of infinitelygenerated free nilpotent groups and so on.4) Every nonidentity algebraically closed group G is generated by the conjugacyclass of any nontrivial element in two steps [12, Cor. 2]. Accordingly, G is a groupof finite C -width.5) Formally, if a given group has only finitely many conjugacy classes, thenthis group is a group of finite C -width.6) Evidently, any homomorphic image (any quotient group) of a group of finite C -width is also a group of finite C -width.Let G be a group. The commutator subgroup of a subgroup H of G will bedenoted by H ′ . Our next example is2.1.
The commutator subgroup F ′ of Thompson’s group F . By the defi-nition, subgroups H , H of a group G are said to be commuting if [ H , H ] = { } , that is, if every element of H commutes with every element of H . For convenience’s sake, we reproduce some definitions and results from [6] weare going to use below.
ENERATING GROUPS BY CONJUGATION-INVARIANT SETS 5
Let H G, let f be an element of G, and let m > f m -displaces H, if the subgroups H, f Hf − , f Hf − , . . . , f m Hf − m are pairwise commuting [6, Sect. 2.1]. We shall say that f ω -displaces H if fm -displaces H for all m > . An element x ∈ G is said to be an f -commutator if x is conjugate to a commu-tator of the form [ f, g ] where g ∈ G. Observe that the inverse of an f -commutatoris also an f -commutator. Proposition 2.2. (i).
Suppose that f m -displaces a subgroup H G for some m > . Then every element of H ′ which has commutator length m in H can bewritten as a product of an f -commutator and a commutator of some elements of G (see the proof of Theorem 2.2 (i) in [6]) . (ii). Suppose that f ∈ G ω -displaces a subgroup H of G. The commutatorlength ( in G ) of every element h ∈ H ′ is at most two [6, Lemma 2.2] . (iii). Suppose that f m -displaces a subgroup H of G where m > . Then everycommutator of elements of H is a product of two f -commutators [6, Lemma 2.7] . Proposition 2.3.
Let f be an arbitrary nonidentity element of the commutatorsubgroup F ′ of Thompson’s group F. Then every element of F ′ is a product ofat most six elements of C ( f ) ± where C ( f ) is the conjugacy class of f in F ′ . Consequently, the commutator subgroup F ′ of Thompson’s group F is a group offinite C -width.Proof. We shall work with the group PL ([0 , , one of the standard realizationsof F [4, 7]. So we assume that F = PL ([0 , . The next result is well-known (folklore; see Proposition 8.1 in [1]).
Lemma 2.4.
Every commutator [ a, b ] ∈ F ′ where a, b ∈ F can be written in theform [ a , b ] where a , b are already elements of F ′ . Recall that every element of F ′ acts identically on a certain closed segment[0 , β ] and on a certain closed segment [ γ,
1] where 0 < β < γ < f be a nonidentity element of F ′ . Suppose that 0 < α ∈ [0 , f, that is, the fixed point of f such that theinterval [0 , α ] is fixed by f pointwise, and every open neighbourhood of α haspoints that are not fixed by f. Let, further, α < f, that is, the point of (0 ,
1) such that there are no fixed points of f in the openinterval ( α , α ) . Take a dyadic point α ∈ ( α , α ) ∩ Z [ ] . Then either α < α < f α < f α < . . . < f n α < . . . < α , ENERATING GROUPS BY CONJUGATION-INVARIANT SETS 6 in the case when α < f α, or α < . . . < f n α < . . . < f α < f α < α < α in the case when α > f α, since f is an increasing function.Consider the open interval J with endpoints α and f α. Clearly, the intervals
J, f J, f J, . . . , f m J, . . . (2.1)all containing in the open interval ( α , α ) , are pairwise disjoint.Let H be the subgroup of all members F ′ whose supports are in J. It is easyto see that H is isomorphic to F, sincePL ( J ) ∼ = PL ([0 , F, where J is the closure of J [4, Prop. 1.4.4].Recall that for every n > , the group F ′ acts transitively on the family of n -element ordered tuples of (0 , ∩ Z [1 /
2] [7, Lemma 4.2]. It follows that everyfinite tuple of elements of F ′ can be taken into H ′ by conjugation by a suitableelement of F ′ . As open intervals in (2.1) are pairwise disjoint, we get that f ω -displaces H in F ′ . Using Lemma 2.4 and part (i) of Proposition 2.2, we can write any element h ∈ H ′ as a product of an f -commutator and a commutator of elements of F ′ : h = [ f, b ] c [ b , b ] . (2.2)Conjugating then the elements b , b by an appropriate c ∈ F ′ , we obtain thecommutator [ b c , b c ]of elements of H. By part (iii) of Proposition 2.2, the latter commutator is aproduct of two f -commutators. Then we deduce from (2.2), that h c is a productof three f -commutators. Accordingly, h is a product of three f -commutators,and then every element of F ′ is a product of three f -commutators. Clearly, aproduct of three f -commutators is a product of six elements of C ( f ) ± , where C ( f ) is the conjugacy class of f in F ′ . (cid:3) Remarks 2.5. (i) As it has been demonstrated in the course of the proof of theproposition, every element of F ′ is a product of two commutators (folklore; see[1, Prop. 8.1]). The problem whether every element of F ′ is a commutator isopen.(ii) The group F itself is not a group of finite C -width, for the abelianization F/F ′ ∼ = Z of F is infinite. As for F ′ , it is not a group of finite width, since itswidth relative to the generating set { x n x − n +1 : n = 0 , , , . . . } is infinite. ENERATING GROUPS BY CONJUGATION-INVARIANT SETS 7 (iii). Proposition 2.3 implies the well-known fact of simplicity of the group F ′ . Subgroups of ‘bounded’ automorphisms of relatively free algebras.
Let V be a variety of algebras and let F ∈ V be a free V -algebra of infinite rank κ . Consider a basis B of F and an infinite cardinal l < κ . For every ϕ ∈ Aut( F ) , define the support of ϕ as supp( ϕ ) = { b ∈ B : ϕb = b } . Set Aut l, B ( F ) = { ϕ ∈ Aut( F ) : | supp( ϕ ) | l } . Clearly, G = Aut l, B ( F ) is a subgroup of the automorphism group Aut( F ) of F , and every element of G fixes the ‘most’ of elements of B . Proposition 2.6.
The group G = Aut l, B ( F ) is a group of finite C -width.Proof. Recall that a subset J of an infinite set I is called a moiety of I if | J | = | I \ J | . We term an automorphism of a given relatively free algebra permutational if itfixes setwise a certain basis of this algebra.Let M be a relatively free algebra with an infinite basis X . Let Y be a moietyof X . Consider a permutational automorphism π of M which acts on Y as aninvolution without fixed points and which fixes pointwise all elements of X \ Y . As it has been demonstrated in the proof of part (i) of Theorem 1.5 in [18], everyautomorphism M which fixes the subalgebra hY i setwise and takes every elementof the set X \ Y to itself is a product of at most four conjugates of π. Choose a subset C of the basis B of cardinality l and a moiety C of C . Considera permutational automorphism π ∗ ∈ G which • fixes the set B \ C pointwise; • acts on the set C as an involution without fixed points.Let H be a subgroup of G consisting of all automorphisms of F preserving thesubalgebra hCi generated by C as a set and taking to themselves all elements of B \ C . Clearly, for every σ ∈ G there is a permutational automorphism ρ ∈ G such that the conjugate σ of σ by ρ is in H, fixes the subalgebra hC i setwise,and fixes the set B \ C pointwise.Then we obtain, as a corollary of the above-quoted result from [18], that σ isa product of at most four conjugates of π ∗ in H ∼ = Aut( hCi ) . Consequently, σ is aproduct of at most four conjugates of π ∗ in G, whence G = C ( π ∗ ) where C ( π ∗ )is the conjugacy class of π ∗ in G. Thus the group G is generated by the class C ( π ∗ ) in at most of four steps, andthen it is a group of finite C -width by Lemma 2.1. (cid:3) ENERATING GROUPS BY CONJUGATION-INVARIANT SETS 8 C -width and group-theoretic constructions Group extensions.
The proof of the fact that the class of all groups offinite width is closed under formation of group extensions given by Bergman in[5] is based upon the following statement.
Lemma 3.1. [5, Lemma 7]
Let U = U − be a symmetric generating set and let H be a subgroup of given group G such that for some natural number nHg ∩ U n = ∅ for all g ∈ G. Then H is generated by those elements G that have length n + 1 relative to U : H = h H ∩ U n +1 i . Proposition 3.2.
Let G be a group and let H be a normal subgroup of G suchthat both the subgroup H and the quotient group G/H are groups of finite C -width.Then G is also a group of finite C -width.Proof. Let U be a symmetric, conjugation-invariant generating set of G and let f : G → G/H be the natural homomorphism from G onto G/H.
Clearly, f ( U ) isalso a conjugation-invariant generating set of the group f ( G ) = G/H.
But thenwid( f ( G ) , f ( U )) = n < ∞ , and we are in the conditions of Lemma 3.1. Evidently, H ∩ U n +1 is a conjugation-invariant generating set of H :( H ∩ U n +1 ) h = H h ∩ ( U n +1 ) h = H ∩ U n +1 ( h ∈ H ) . Hence, by the conditions,wid(
H, H ∩ U n +1 ) = m < ∞ , whencewid( G, U ) wid( f ( G ) , f ( U )) + wid( H, U ) n + (2 n + 1) m < ∞ . (cid:3) We then obtain as a corollary that the class of all groups of finite C -width isclosed under formation of group extensions, and, in particular, it is closed underformation of direct products and under formation of group extensions by finitegroups. ENERATING GROUPS BY CONJUGATION-INVARIANT SETS 9
Free products.
Essentially, the following result states that the only freeproduct of nonidentity groups which is a group of finite C -width is, up to anisomorphism, the free product Z ∗ Z . Proposition 3.3.
The free product Q ∗ i ∈ I G i of a family { G i : i ∈ I } of noniden-tity groups is a group of finite C -width if and only if | I | = 2 and both groupsparticipating in the free product are of order two.Proof. ( ⇐ ) . Consider groups A = h a i and B = h b i , both isomorphic to the group Z . Let w ∈ A ∗ B be a reduced word in letters a, b which begins with a. Then w = ( ab ) k a, or w = ( ab ) k for a suitable natural number k. It is easy to see that each word of the form( ab ) k a is conjugate either to a, or to b. Let k > . Then( ab ) k = ( ab ) k − ab = ( ab ) k − a · b and the element ( ab ) k is a product of at most two conjugates of elements of { a, b } . The argument in the case when w begins with b is similar. Therefore the group A ∗ B is generated in two steps by the union of the conjugacy class of a and theconjugacy class of b. Apply Lemma 2.1 to complete the proof.( ⇒ ) . Recall that a map ∆ : H → Z from a given group H into Z is called a quasi-homomorphism if there is a constant C such that∆( ab ) ∆( a ) + ∆( b ) + C ( a, b ∈ H ) . Fix a family { G i : i ∈ I } of nonidentity groups, and let G denote the freeproduct Q ∗ i ∈ I G i . An element p of the free product G is called a palindrome if thereduced word representing p (whose syllables are nonidentity elements of factors G i ) is read the same way forwards and backwards. Thus if p ∈ G and p = v . . . v n where v k are nonidentity elements of free factors G i such that the elements v m , v m +1 lie in distinct free factors for all m, then p is a palindrome if and only if v . . . v n = v n . . . v . In [3] quasi-homomorphisms have been used to show that all free products ofnonidentity groups that are not isomorphic to the group Z ∗ Z have infinitewidth relative to the (generating) set of all palindromes, or, in other words,infinite palindromic width.It particular, it is proved in [3] that if the free product G = Q ∗ i ∈ I G i where | I | > G i has an element of order > : G → Z with the following properties: ENERATING GROUPS BY CONJUGATION-INVARIANT SETS 10 (a) ∆ ( ab ) ∆ ( a ) + ∆ ( b ) + 9 for all a, b ∈ G ;(b) the value ∆ at any palindrome of G is at most 2;(c) ∆ is not bounded from above.(see [3, pp. 203–204]).1) Suppose then that our free product G has the property (3.1). The readercan verify quite easily that it follows from the definition of ∆ in [3] that∆ ( aba − ) ∆ ( b ) + 9 ( a, b ∈ G ) . By (b), this implies that the value of ∆ at any conjugate of a palindrome in G isat most 11 . Now, were the width of G relative to the family of all conjugatesof palindromes finite, the quasi-homomorphism ∆ would be bounded on G, contradicting (c). Therefore every free product of nonidentity groups with (3.1)is not a group of finite C -width.2) Now let G = Z ∗ Z not satisfy (3.1). Then G is a free product of nontrivialabelian groups of exponent two. It is easy to see that in any free product ofabelian groups of exponent two, a conjugate of a palindrome is a palindrome,too. Thus the set of all palindromes of G is invariant under all conjugations. Aswe have mentioned above, the palindromic width of G is infinite, and hence G isnot a group of finite C -width. (cid:3) Functions similar to length functions and cofinalities.
Basing on theideas from the paper [5] by Bergman, several authors obtained a number of nec-essary and sufficient conditions for all functions L : G → R on a given group G such that L − (0) = { } , (3.2) L ( g − ) = L ( g ) ( g ∈ G ) ,L ( gh ) L ( g ) + L ( h ) ( g, h ∈ G ) . to be bounded from above. For instance, this takes place if and only if G isa group of finite width and the cofinality cf( G ) is uncountable [9, 10]; anothersuch criterion states that every action of G by isometries on a metric space hasbounded orbits [8].By the definition, the cofinality cf( G ) of an infinitely generated group G is theleast cardinal λ such that G can be written as the union of a chain of cardinality λ of its proper subgroups (observe that no finitely generated group can be written asthe union of a chain of proper subgroups). In the case when cf( G ) > ℵ , a group G is said to be a group of uncountable cofinality. For instance, the symmetricgroup Sym( X ) of an infinite set X is a group of uncountable cofinality, sincecf(Sym( X )) > | X | [13, 5]. ENERATING GROUPS BY CONJUGATION-INVARIANT SETS 11
In accordance with the terminology introduced in [10], if G is a group of finitewidth and of uncountable cofinality, G is called a group of strong uncountablecofinality. As we mentioned in the introduction, the paper [6] contains a number of resultson functions (termed norms in [6]) L : G → R on groups satisfying the conditions(3.2) and taking constant values on conjugacy classes, that is, satisfying theadditional condition L ( ghg − ) = L ( h ) ( g, h ∈ G ) . We shall provide below some necessary and sufficient conditions for all norms ona given group G to be bounded from above. Our conditions are in fact naturally-weakened versions of conditions, equivalent to strong uncountable cofinality, thatcan be found in papers [8, 10, 15]. Proposition 3.4.
Let G be a group. Then the following are equivalent: (i). G is a group of finite C -width and every exhaustive chain ( N k ) N N . . . N k . . . G of normal subgroups of G ( every increasing chain of normal subgroups whoseunion is G ) terminates after finitely many steps; (ii). Every exhaustive chain ( U k ) U ⊆ U ⊆ . . . ⊆ U k ⊆ . . . ⊆ G of subsets of G such that for every i ∈ N • U i closed under taking inverses; • U i is conjugation-invariant; • the product U i U i is contained in a suitable U k terminates after finitely many steps; (iii). Orbits of every action of G by isometries on a metric space h M, d i suchthat d ( a, ghg − a ) = d ( a, ha ) ( a ∈ M ; g, h ∈ G ) have bounded diameters; (iv). Every function L : G → R taking constant values on conjugacy classes of G and such that • L ( g ) = 0 if and only if g = 1; • L ( gh ) L ( g ) + L ( h ) for all g, h ∈ G is bounded from above. ENERATING GROUPS BY CONJUGATION-INVARIANT SETS 12
Proof. (i) ⇒ (ii). Clearly, the chain of subgroups of G generated by sets U i , h U i h U i . . . h U k i . . . G, is an exhaustive chain of normal subgroups of G. Then G = h U j i for a suitablenatural number j, and hence U j is a symmetric, conjugation-invariant generatingset of G. As G is a group of finite C -width, G = U sj . By the conditions on thechain ( U k ) , the power U sj is contained in some U m for an appropriate m ∈ N , whence U m = G. (ii) ⇒ (iii). Let a be an arbitrary element of a metric space M satisfying (iii).Set U n = { g ∈ G : d ( a, ga ) n } ( n ∈ N ) . It follows from (iii) that every U n is conjugation-invariant. Let g, h ∈ U n . Thenwe have that d ( a, gha ) d ( a, ga ) + d ( ga, gha ) = d ( a, ga ) + d ( a, ha ) n + n = 2 n. Consequently, U n U n ⊆ U n . As the chain ( U n ) terminates, we get that G = U m for some m ∈ N . Hence d ( a, ga ) m for all g ∈ G. Thus the diameter of the orbit { ga : g ∈ G } of a ∈ M is at most2 m. (iii) ⇒ (iv). Let a, b ∈ G. Set d ( a, b ) = L ( ab − ) . It is easy to see that d is a metric on G satisfying the conditions in (iii) for theleft action G on itself. Indeed, we have that d ( a, b ) = 0 ⇐⇒ L ( ab − ) = 0 ⇐⇒ ab − = 1 ⇐⇒ a = b,d ( a, b ) = L ( ab − ) = L ( ba − ) = d ( b, a ) ,d ( a, b ) = L ( ab − ) = L ( ac − · cb − ) L ( ac − ) + L ( cb − ) = d ( a, c ) + d ( c, b ) ,d ( ga, gb ) = L ( gab − g − ) = L ( ab − ) = d ( a, b ) ,d ( a, ghg − a ) = L ( ghg − ) = L ( h ) = d ( a, ha )for all a, b, g, h ∈ G. Then the orbit of 1 ∈ G under the left action of G on itselfhas a bounded diameter m ∈ N , or L ( g ) = d ( g , m ( g ∈ G ) . (iv) ⇒ (i). Let S = S − be a symmetric, conjugation-invariant generating setof G. Then the function L ( g ) = | g | S ( g ∈ G ) ENERATING GROUPS BY CONJUGATION-INVARIANT SETS 13 that is, the length function with regard to S, which meets all conditions mentionedin (iv), must be bounded from above by some natural number m. Accordingly, G = S m . Let further ( N k ) be an exhaustive chain of normal subgroups of G. For every g ∈ G set L ( g ) = min { k ∈ N : g ∈ N k } ( g ∈ G ) . It is readily seen that L satisfies all conditions in (iv). For example, L ( ab ) max( L ( a ) , L ( b )) L ( a ) + L ( b )for all a, b ∈ G. One again concludes that L is bounded from above by a certainnatural number m, whence G = N m . (cid:3) Remark 3.5.
In the case when G is a simple group, Proposition 3.4 provides acriterion of finiteness of C -width of G. In the conclusion of the section we shall discuss some notion which generalizesboth the notion of a group of finite width and the notion of a group of finite C -width. Let G be a group and let Σ Aut( G ) be a subgroup of the automorphismgroup of G. We say that G has finite Σ -width if G has finite width with respectto all Σ-invariant generating sets. Clearly, the case when Σ = { id } correspondsto the notion of a group of finite width, and the case when Σ = Inn( G ) to thatone of finite C -width.Intuitively, the greater the (setwise) stabilizer in Aut( G ) of a given set S ofgenerators of G , the more ‘massive’ S appears to be with the ‘point of view’ ofthe automorphism group of G. Thus if G has finite Σ-width in the case whenΣ = Aut( G ) , it has finite width with regard to all ‘most massive’ generating sets.One can, as we did in Proposition 3.4, add to the condition of finiteness ofΣ-width the condition of termination of all exhaustive chains of Σ-invariant sub-groups of G. Modifying then the formulation of Proposition 3.4 accordingly, onecan obtain necessary and sufficient conditions for G to have finite Σ-width and,simultaneously, to satisfy the condition of termination of all exhaustive chains ofΣ-invariant subgroups. For instance, the analogue of the part (iii) of Proposition3.4 is as follows: every action of G by isometries on a metric space h M, d i forwhich d ( σ ( g ) a, a ) = d ( ga, a ) , ( a ∈ M, σ ∈ Σ , g ∈ G )has bounded orbits.The case when Σ is equal to the full automorphism group Aut( G ) of G seemsto be quite interesting. Simplifying the terminology somewhat, we say that G has finite Aut -width if G has finite width with respect to every generating setwhich is invariant under all automorphisms of G. ENERATING GROUPS BY CONJUGATION-INVARIANT SETS 14
Our final result shows that the class of all groups of finite Aut-width does nothave some attractive properties that its counterparts, the classes of all groups offinite width and all groups of finite of C -width, have (in particular, this class isnot closed under homomorphic images). Nevertheless, the property of having/nothaving finite Aut-width can be used to distinguish between the isomorphism typesof groups. Proposition 3.6.
A free group F is a group of finite Aut -width if and only if itsrank is infinite.Proof.
Suppose that F is of finite rank. Consider the (generating) set P of allprimitive elements of F. Clearly, P is invariant under all automorphisms of F, but the width of F relative to P is infinite [2, Th. 2.1].Now let F be of infinite rank. This time, the width of F with regard to the set ofall primitive elements is two [2, Th. 2.1]. Consider a symmetric generating set S of F which is invariant under all automorphisms of F. Then a certain power S k of S contains a primitive element p ∈ F. As S is invariant under automorphisms of F, the said power of S contains all primitive elements. Consequently, F = S k . (cid:3) References [1] T. Altinel, A. Muranov, Interpretation de l’arithmetique dans certains groupes de permu-tations affines par morceaux d’un intervalle.
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E-mail address : [email protected] Vladimir Vershinin, Department of Mathematics, University Montpellier II,Place Eug`ene Bataillon, 34095 Montpellier Cedex 5, France
E-mail address : [email protected] Vladimir Tolstykh, Department of Mathematics, Yeditepe University, 34755Kayıs¸da˘gı, Istanbul, Turkey
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