Lower central subgroups of a free group and its subgroup
aa r X i v : . [ m a t h . G T ] J u l Lower central subgroups of a free group and itssubgroup
Minkyoung Song
Abstract.
For a given free group F of arbitrary rank (possibly infinite), and its subgroup G ,we address the question whether a lower central subgroup of G can contain a lower centralsubgroup of F . We show that the answer is no if G does not normally generate F . The questioncomes from a study of Hirzebruch-type invariants from iterated p -covers for 3-dimensionalhomology cylinders.
1. Introduction
For a group G , denote by G m the m th term of the lower central series of G , definedinductively by G = G , G m +1 = [ G m , G ] for each m ≥ F is a free group and G its subgroup, then it is obvious that G m is containedin F m for every m ≥
1. In this paper, we investigate the converse relation: whethersome F m is contained in some G k . Note that T m F m = 0. For k = 1, if G is a normalsubgroup of F with abelian F/G , then G contains F m for every m ≥
2. We can askif a subgroup G satisfies G ⊇ F m for a certain large m . As an answer, we prove thefollowing result: Theorem 1.1.
Let F be a free group and G a subgroup of F whose normal closure isnot F . Then G never contains F m for any m ∈ N . This starts from a study of structures of geometric objects. Let Σ g,n be a compactoriented surface of genus g with n boundary components. A homology cylinder overΣ g,n is defined as a homology cobordism between two copies of Σ g,n . The set H g,n of homology cobordism classes of homology cylinders becomes a group under juxtapo-sition. The group was introduced as an enlargement of the mapping class group byGaroufalidis and Levine [GL05, Lev01]. It is also a generalization of the concordancegroup of framed string links.In [So16], the author studied the structure of H g,n by defining extended Milnorinvariants and Hirzebruch-type invariants for homology cylinders. Throughout thispaper, p denotes a prime number. Hirzebruch-type intersection form defects associatedto p r -fold covers are defined by Cha in [Cha10] to study homology cobordism of closed 3-manifolds and concordance of links. Let d be a power of p . For a CW-complex X , a pairof a cover ˜ X obtained by taking p -covers repeatedly and a homomorphism π ( ˜ X ) → Z d is called a ( Z d -valued ) p -structure for X . Here, a p -cover means a cover of p -powerdegree. The invariant of a p -structure for a 3-manifold is the difference between theWitt classes of the Q ( ζ d )-valued intersection form and the ordinary intersection formof a 4-manifold bounded by ˜ X over Z d , where ζ d = exp(2 π √− /d ). This lives in theWitt group L ( Q ( ζ d )) of nonsingular hermitian forms over Q ( ζ d ). The invariants give rise to invariants of a subgroup of string link concordance group,consisting of b F -string links [Cha09]. We refer [Cha09, p.897] for the definition of b F -string link. Remark that b F -(string) links form the largest known class of (string) linkswith vanishing Milnor invariants; it is a big open problem in link theory whether all(string) links with vanishing Milnor invariants are b F -(string) links. It turned out thatthe Hirzebruch-type invariants are homomorphisms on the subgroup of b F -string links.In [So16], a Hirzebruch-type invariant λ T is defined for homology cylinders with a p -structure T for Σ g,n , or equivalently for W g + n − S . The p -structures are classifiedin [So16]; Let ( ˜ X, φ : π ( ˜ X ) → Z d ) be a p -structure for X . For the cover ˆ X inducedby φ , if π ˆ X ⊇ ( π X ) m , the p -structure is said to be of order m . Every p -structureof a finite CW-complex is of order m for some finite m ; For the proof, see [So16,Lemma 5.3]. We revealed that when the invariant is defined; For a p -structure T oforder m , the invariant λ T is defined for (the homology cobordism class of) a homologycylinder if and only if the homology cylinder has vanishing extended Milnor invariantsof length m . Let H g,n ( m ) be the subgroup of H g,n consisting of homology cylinderswith vanishing extended Milnor invariants of length m in H g,n . For a p -structure T for Σ g,n of order m , the Hirzebruch-type invariant λ T : H g,n ( m ) −→ L ( Q ( ζ d ))is well-defined. A sufficient condition that λ T is additive is given in [So16, Theo-rem 5.12]. It follows that λ T is a homomorphism on T m H g,n ( m ) for any p -structure T .Using homomorphisms λ T , it turned out that the abelianization of T m H g,n ( m ) con-tains a subgroup isomorphic to Z ∞ if b (Σ g,n ) = 2 g + n − > m such that the λ T are homomorphisms on H g,n ( m ), then we will obtainthat H ( H g,n ( m )) also contains a subgroup isomorphic to Z ∞ . To find λ T which isa homomorphism on H g,n ( m ), the author extracted the following from the sufficientcondition. Proposition 1.2. [So16, Corollary 5.13]
Let
Σ = Σ g,n . Suppose T = ( ˜Σ , π ˜Σ → Z d ) is a p -structure for Σ of order m . If ( π ˆΣ) ⊇ ( π Σ) m for the Z d -cover ˆΣ of ˜Σ then T gives a homomorphism λ T : H g,n ( m ) → L ( Q ( ζ d )) . This naturally poses the problem to find a p -structure T for Σ satisfying the as-sumption of the proposition. The problem can be interpreted algebraically as follows: Problem.
Suppose F is a finitely generated free group. Find a proper subgroup G of F such that there is an ascending chain G = F ( k ) ⊳ F ( k − · · · ⊳ F (1) ⊳ F (0) = F witheach F ( i ) /F ( i +1) an abelian p -group and G ⊇ F m for some m .We can simplify the problem as follows: Problem. (simple version) Suppose F is a finitely generated free group. Find a propernormal subgroup G such that F/G is abelian and G ⊇ F m for some m .This is equivalent to the following geometric problem which is the core of the originalproblem: Problem.
Let X be a CW-complex with π X free. Find an abelian cover ˜ X of X such that the natural map π ˜ X/ ( π ˜ X ) m → H ( ˜ X ) factors through π ˜ X/ ( π X ) m forsome m ≥ λ T on H g,n ( m ) since OWER CENTRAL SUBGROUPS OF A FREE GROUP AND ITS SUBGROUP 3
Proposition 1.2 follows from only a sufficient condition for λ T to be additive in [So16,Theorem 5.12].Extending the domain of λ T as a homomorphism may help study the mapping classgroups of surfaces. The restriction of H g,n ( m ) on the mapping class group is the Johnson filtration M g,n [ m ] := Ker {M g,n → Aut(
F/F m ) } . In other words, H g,n ( m ) ∩M g,n = M g,n [ m ]. The subgroups M g,n [2] and M g,n [3] are well known as the Torelligroup and the
Johnson kernel , respectively. In 1938, Dehn proved that M g,n is finitelygenerated [Den38]. In 1983, Johnson proved that M g, [2] and its quotient M g, [2]are also finitely generated for g ≥
3, but it is discovered that M , [2] and M , [2] = M , [3] are infinitely generated by McCullough and Miller [MM86] in 1986. Thereby,the question whether M g,n [3] is finitely generated for g ≥ n = 0 ,
1, Ershov and He [EH17] showed that M g,n [3] is finitely generated if g ≥
12 and H ( M g,n [ m ]) is also finitely generated if m ≥ , g ≥ m −
12. Church, Ershov and Putman proved that also for n = 0 , M g,n [3]is finitely generated if g ≥ M g,n [ m ] is finitely generated if m ≥ , g ≥ m − M g,n [ m ] is finitely generated for general g and n . TheHirzebruch-type invariants may be used to prove that the abelianization is infinitelygenerated if we find a homomorphism λ T on the higher order Johnson subgroup. Acknowledgements
The author thanks Jae Choon Cha for helpful comments. The work was supported byNRF grant 2011-0030044 (SRC-GAIA).
2. Non-existence of subgroups
We denote [ x, y ] := xy ¯ x ¯ y where ¯ x means x − . Theorem 2.1.
Suppose F is a finitely generated free group. Then there is no normalsubgroup of F of prime index whose commutator subgroup contains a term of the lowercentral series of F .Proof. Suppose there is an index p normal subgroup G of F such that the commutatorsubgroup [ G, G ] contains F m for some m ∈ N . Then G can be considered as the kernelof a surjective homomorphism F ։ Z p .It is enough to show that if F = h x, y i and G = Ker { F f −→ Z p } where f ( x ) =1 , f ( y ) = 0, then G + F m for all m .Let ω n := [ · · · [[ x, y ] , x ] , . . . , x ] = [ x, y, x, . . . , x | {z } n times ] ∈ F n +2 for n ≥
0. We claim that ω n / ∈ G for every n ∈ N . Since ω n is an element of G , our claim is equivalent that[ ω n ] = 0 in G/G = H ( G ).The subgroup G = hh x p , y ii F = h x p , y, xy ¯ x, x y ¯ x , . . . , x p − y ¯ x p − i . Let a := x p and b k := x k − y ¯ x k − for k = 1 , . . . , p , then G = h a, b , . . . , b p i . Denote by S the freegenerating set { a, b , . . . , b p } .For ω ∈ G and k = 1 , . . . , p , let P k ( ω ) be the sum of the powers of b k in ω as a wordexpressed in S . In other words, P k ( ω ) is the power of [ b k ] in [ ω ] ∈ H ( G ). We notethat(1) xa ¯ x = a, xb ¯ x = b , . . . , xb p − ¯ x = b p , xb p ¯ x = ab ¯ a. OWER CENTRAL SUBGROUPS OF A FREE GROUP AND ITS SUBGROUP 4
Thus, conjugating any element of G by x preserves the sum of powers of a in a wordin S .Since a does not appear in the reduced word of ω = xy ¯ x ¯ y = b ¯ b , [ ω n ] = 0 ∈ H ( G )if and only if P k ( ω n ) = 0 for all k . We observe P ( ω ) = − , P ( ω ) = 1 , P k ( ω ) = 0for k ≥
3, and P k ( ω n +1 ) = P k ([ ω n , x ]) = P k ( ω n ) + P k ( x ¯ ω n ¯ x ) = P k ( ω n ) − P k ( xω n ¯ x ) = P k ( ω n ) − P k − ( ω n ). The last equality comes from (1). Hence we obtain P ( ω n ) P ( ω n ) P ( ω n )... P p ( ω n ) | {z } v n = − − − − | {z } A n − | {z } v Let us calculate the eigenvalues of A . Since det( A − λI ) = (1 − λ ) p −
1, the eigenvalues λ j of A are 1 − ζ j where ζ is the p -th root of unity e πi/p and j = 1 , . . . , p . Thecorresponding eigenvector x j to the eigenvalue λ j is ζ ( p − j ... ζ j ζ j . Since the eigenvalues λ j are all distinct, x j are linearly independent. Thus, v can beexpressed as a linear combination of x j . Let v = P pi =1 α j x j . Note that α j is nonzerofor some j = p since v = α x p for any α . Therefore, v n = A n v = P pi =1 α j λ nj x j isnonzero for any n ≥
1. In conclusion, ω n is not an element of G , and it implies that G does not contain any F m . (cid:3) Note that prime index does not guarantee normality. For instance, there is a non-normal subgroup h a, b , ba b, babab i of index 3 in Z ∗ Z = h a, b i .In fact, the same argument holds not only for p prime, but also when p is replacedby an arbitrary integer >
1. Hence the theorem also holds not only for index p normalsubgroups but also for normal subgroups with finite cyclic factor groups. Moreover,we can extend Theorem 2.1 as follows: Corollary 2.2.
Let F be a (possibly infinitely generated) free group. Suppose G is asubgroup of F such that there are H and K with G ≤ K ⊳ H ≤ F , a nontrivial abelianfactor group H/K . Then G does not contain F m for any m ∈ N .Proof. First we generalize Theorem 2.1 to a free group of arbitrary rank. Let G be anormal subgroup of index p where p is a prime. We can assume that { x i | i ∈ I } isa free generating set of F with an index set I ∋ , G = Ker { f : F ։ Z p } with f ( x ) = 1, f ( x j ) = 0 for j = 1 ∈ I . Suppose G ⊇ F m for some m . Let H = h x , x i ,a subgroup of F . Then, H ∩ G = Ker { f | H : H ։ Z p } is an index p normal subgroupof H . But, ( H ∩ G ) = H ∩ G ⊇ H ∩ F m ⊇ H m . It contradicts Theorem 2.1. OWER CENTRAL SUBGROUPS OF A FREE GROUP AND ITS SUBGROUP 5
Now let us extend G to a subgroup of F with G ≤ K ⊳ H ≤ F and nontrivial abelian H/K . Suppose G ⊇ F m for some m ∈ N . Then, K ⊇ G ⊇ F m ⊇ H m . There is aprime index normal subgroup K ′ of H which contains K since there is an epimorphismof H/K onto a cyclic group of prime order. We have ( K ′ ) ⊇ K ⊇ H m , which is acontradiction. (cid:3) For instance, if
F/G is the alternating group A , it has abelian subgroups isomorphicto Z , Z , Z , so G satisfies the hypothesis of the above corollary.Lastly, we give a proof of Theorem 1.1 stated in the introduction: Proof of Theorem 1.1.
Let K be the normal closure of G . Every nontrivial group hasa nontrivial abelian subgroup, so there is a nontrivial abelian subgroup H/K of F/K .Then G ≤ K ⊳ H ≤ F satisfies the hypothesis of Corollary 2.2. Consequently, theconclusion of the corollary holds for every subgroup whose normal closure is not F .Hence we obtain Theorem 1.1. (cid:3) References [Cha09] J. C. Cha,
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