On the divisibility of $#\Hom(Γ,G)$ by $|G|
aa r X i v : . [ m a t h . G R ] M a y On the divisibility of Γ , G ) by | G | Cameron Gordon
University of Texas at Austin [email protected]
Fernando Rodriguez-Villegas
University of Texas at Austin [email protected]
June 23, 2018
Abstract
We extend and reformulate a result of Solomon on the divisibility of the title. We show, for example,that if Γ is a finitely generated group, then | G | divides Γ , G ) for every finite group G if and onlyif Γ has infinite abelianization. As a consequence we obtain some arithmetic properties of the number ofsubgroups of a given index in such a group Γ . Let G be a finite group. For any non-negative integer g and a fixed z ∈ G consider the set U z : = { ( x , y , . . . , x g , y g ) ∈ G g | [ x , y ] · · · [ x g , y g ] = z } , (1.0.1)where [ x , y ] : = xyx − y − . We have [3] U z = X χ | G | χ (1) ! g − χ ( z ) , (1.0.2)where the sum is over all irreducible characters of G (formulas of this sort were already known to Frobe-nius).In particular, for z = U = | G | X χ | G | χ (1) ! g − . (1.0.3)Since χ (1) divides | G | for every χ , it follows that | G | divides U when g >
0. The purpose of this note isto give a more direct proof of a generalization of this observation.We may interpret U as the set Hom( π ( Σ g ) , G ) of homomorphisms of the fundamental group of aRiemann surface Σ g of genus g to G . The question is then the following. Given a finitely generated group Γ , say, when do we have that | G | divides Hom( Γ , G ) for every finite group G ? It turns out that the answer isquite simple: precisely when Γ has an infinite abelianization (see Corollary 3.3).As it happens, the answer to our question is buried in a paper of Solomon’s [4], which was the mainmotivation for the present note. Solomon’s proof assumes that Γ has positive deficiency (i.e., has a presen-tation with strictly more generators than relations), but it is easy to see that it can be made to apply more1generally to any group with infinite abelianization. However, his proof is lengthy and somewhat hard toread; the proof we give is short and, we hope, more perspicuous. Solomon deduces his result from a moregeneral statement about homomorphisms from a free group to G that send specified elements into specifiedconjugacy classes. As an immediate corollary to our theorem we get a generalization of this in which thefree group is replaced by any group whose abelianization is infinite. G -torsors Let G be a finite group. A G -set X is a set X on which G acts. We denote by X / G the set of orbits under theaction. Let G be the bundle of groups Stab G ( x ) on X . A G -torsor of X is a surjective map of sets π : Y → X ,for some set Y , such that each fiber Y x is a principal homogeneous space for Stab G ( x ). In other words, thereis an action of Stab G ( x ) on Y x for every x ∈ X such that for any fixed y ∈ Y x the mapStab G ( x ) → Y x s sy is a bijection. It is worth stressing that we do not require a global action of G on Y inducing the action ofStab G ( x ) on Y x .We will write | · | for the order of a group. Lemma 2.1.
Let X be a G-set and π : Y → X a G -torsor. Then(i) X is finite if and only if Y is finite.(ii) In this case, Y / | G | = X / G ) . In particular Y / | G | is an integer or, equivalently, | G | divides Y.Proof.
The first assertion (i) is clear since G is finite. As for (ii) note that Y = X x ∈ X Y x = X x ∈ X | Stab G ( x ) | . Since | Stab G ( x ) | is constant on orbits in X / G we can group terms in the sum by orbits. The contribution tothe sum of an orbit [ x ] is then x ] | Stab G ( x ) | = | G | . Hence Y = X [ x ] | G | = | G | · X / G ) . and the lemma is proved. (cid:3) Let Γ be a group. The set X : = Hom( Γ , G ), where G is a finite group, is then finite if Γ is a finitelygenerated. The group G acts on X by conjugation. The quotient Γ , G ) / | G | that we are interested incan be thought of as a weighted count of homomorphisms from Γ to G . Namely,1 | G | Γ , G ) = X [ φ ] | Stab G ( φ ) | , where [ φ ] runs through the G -orbits of Hom( Γ , G ).We also have an action of Aut( Γ ) on X by φ σ ( γ ) : = φ ( σ − γ ) , σ ∈ Aut( Γ ) , φ ∈ X , γ ∈ Γ . Fix σ ∈ Aut( Γ ) and let X σ : = Hom σ ( Γ , G ) ⊆ Hom( Γ , G ) be the subset of φ ∈ X which are fixed by σ up to conjugation by G . The two actions, of G and Aut( Γ ), on X clearly commute. In particular, the actionof Aut( Γ ) passes to the quotient X / G and X σ is a G -set.Let Γ ⋊ σ Z be the semidirect product of Γ and Z induced by the map Z → Aut( Γ ) that takes 1 to σ . Ahomomorphism Φ ∈ Hom( Γ ⋊ σ Z , G ) is uniquely determined by the pair ( φ, g ), where φ : = Φ | Γ and g ∈ G is the image of (1 , ∈ Γ ⋊ σ Z . The necessary and su ffi cient condition for ( φ, g ) to arise from a Φ in thisway is φ σ ( γ ) = g φ ( γ ) g − , γ ∈ Γ . In particular, φ ∈ Hom σ ( Γ , G ). It follows that if we fix one pair ( φ, g ) then the set of all other pairs ( φ, g ′ )are given by setting g ′ = gs with s ∈ Stab G ( φ ). We may hence define an action of Stab G ( φ ) on these pairsby setting s · ( φ, g ) : = ( φ, gs − ) This proves the following. Proposition 3.1.
With the above notation for any finite group G the map
Hom( Γ ⋊ σ Z , G ) → Hom σ ( Γ , G ) given by restriction is a G -torsor of Hom σ ( Γ , G ) . Combining Proposition 3.1 with Lemma 2.1 we obtain the following corollary
Corollary 3.2. If Γ ⋊ σ Z is finitely generated then(i) Hom σ ( Γ , G ) is finite,(ii) moreover, | G | Γ ⋊ σ Z , G ) = σ ( Γ , G ) / G ) . We should point out that for a finitely generated group ˜ Γ to be isomorphic to a Γ ⋊ σ Z for some Γ and σ is equivalent to having infinite abelianization. If ˜ Γ has finite abelianization A then picking, say, G = Z / pZ with p a prime not dividing | A | we see that | G | does not always divide Γ , G ). We have proved thefollowing. Corollary 3.3.
A finitely generated group ˜ Γ has the property that | G | divides Γ , G ) for every finitegroup G if and only if it has infinite abelianization. We can extend the previous results as follows. Let C i be an indexed collection of (not necessarilydistinct) conjugacy classes in G , and let S i be a collection of subsets of Γ . Let Hom ′ ( Γ ⋊ σ Z , G ) ⊆ Hom Γ ⋊ σ Z , G ) consist of those homomorphisms Φ such that Φ ( S i ) ⊆ C i for all i . Define Hom ′ σ ( Γ , G ) ⊆ Hom σ ( Γ , G )similarly and let π ′ be the restriction of π to Hom ′ ( Γ ⋊ σ Z , G ). Theorem 3.4.
The map π ′ takes Hom ′ ( Γ ⋊ σ Z , G ) to Hom ′ σ ( Γ , G ) and π ′ : Hom ′ ( Γ ⋊ σ Z , G ) → Hom ′ σ ( Γ , G ) is a G -torsor of Hom ′ σ ( Γ , G ) .Proof. Note that Hom ′ σ ( Γ , G ) is indeed a sub- G -set of Hom σ ( Γ , G ) since the C i are conjugacy classes andHom ′ ( Γ ⋊ σ Z , G ) = π − (Hom ′ σ ( Γ , G )). Now the claim follows from Proposition 3.1. (cid:3) As before we obtain
Corollary 3.5.
Let Γ ⋊ σ Z be finitely generated. Then | G | ′ ( Γ ⋊ σ Z , G ) = ′ σ ( Γ , G ) / G ) . Remark . The fact that | G | divides ′ ( Γ ⋊ σ Z , G ) when Γ ⋊ σ Z is a free group of rank n , the conjugacyclasses C i are indexed by 1 , , ..., m with m < n , and each S i is a singleton is the main result of [4]. We illustrate some of the issues concerning the ratio ′ ( ˜ Γ , G ) / | G | with a few examples.(1) Take ˜ Γ to be the free group F in k − h x , . . . , x k | x · · · x k = i for some k > S i = { x i } for i = , , . . . , k . Fix a finite group G and conjugacy classes C , . . . , C k of G and letHom ′ ( F , G ) be as above. In general there is no reason to expect ′ ( F , G ) to be divisible by | G | , thoughthe denominator of the quotient is often much smaller than | G | .By a formula extending (1.0.2) we have ′ ( F , G ) = X χ χ (1) | G | k Y i = f χ ( C i ) , (4.0.4)where the sum is over all irreducible characters of G and for a conjugacy class C we define f χ ( C ) : = C χ ( x ) χ (1) , x ∈ C . It is known that f χ ( C ) is an algebraic integer.For example, take G = S n , the symmetric group in n letters, and let C i , for i = , , . . . , k , be theconjugacy class of an n -cycle ρ : = (12 · · · n ). The irreducible characters χ λ of S n are parametrized bypartitions λ of n and χ λ ( ρ ) = ( − r if λ = ( n − r , , . . . ,
1) is the r -th hook and χ λ ( ρ ) = | G | ′ ( F , G ) = n n − X r = (( − r r !( n − r − k − , which is typically not an integer; for example, for k = n − . In fact, for k > n is a prime not dividing k −
1, in which case the denominator equals n . (We thank J. Guntherfor showing us a proof of this fact.)(2) In the previous example take k = G = SL ( F q ) for some q = p r with p > C i , for i = , . . . ,
4, to be the conjugacy class of a diagonal matrix with eigenvalues λ i , λ − i ∈ F × q \ {± } . Then using the known description of the irreducible characters of G we find after somecalculation that in this case 1 | G | ′ ( F , G ) = q + q + + a q q − , where a : = { ( ε , . . . , ε ) ∈ ( ± | λ ε · · · λ ε = } . If q is su ffi ciently large there will be a choice of eigenvalues λ , . . . , λ such that a =
0, in which case ′ ( F , G ) / | G | = q + q + q . In fact, with this choice of generic eigenvalues theaction of G by conjugation on ′ ( F , G ) is actually free explaining the divisibility. (See [1] for moredetails).(3) Consider the group ˜ Γ : = h u , . . . , u n , z , . . . , z k | w ( u , . . . , u n ) · z · · · z k i , where n ≥ w is anarbitrary word in u , . . . , u n that belongs to the commutator of the free group h u , . . . , u n i . Pick a non-trivialhomomorphism ψ : ˜ Γ → Z with ψ ( z i ) = i = , , . . . , k and let Γ : = ker ψ . Take S i = { z i } for i = , , . . . , k and pick arbitrary conjugacy classes C , . . . , C k in G . Then by Corollary 3.5 | G | divides ′ ( ˜ Γ , G ). Concretely, the number of solutions to the equation w ( u , . . . , u n ) · z · · · z k = , u i ∈ G , z i ∈ C i is divisible by | G | for any finite group G . If we take k = w = [ x , y ] · · · , [ x g , y g ] this yields the caseof a Riemann surface (1.0.3) considered in the introduction.(4) Consider the case where ˜ Γ = h x , y | w ( x , y ) i is a two generator, one-relator group. Clearly, ˜ Γ hasinfinite abelianization and hence Γ , G ) / | G | ∈ Z by Corollary 3.3. We can express this quantity interms of irreducible characters of G as follows.1 | G | Γ , G ) = X χ s χ ( w ) χ (1) , where for an irreducible character χ of Gs χ ( w ) : = | G | X x , y ∈ G χ ( w ( x , y ))(see [3]). For example, if w ( x , y ) = xyx − y − then s χ ( w ) = χ (1) − . We may wonder if it is always the casethat s χ ( w ) χ (1) ∈ Z . This turns out not to be true. In fact, the numbers s χ ( w ) χ (1) appear to be fairly arbitraryalgebraic numbers in general.Consider for example, w = x y x − y − and G = PSL ( F ). Then the values of s χ ( w ) χ (1) for thedi ff erent irreducible characters (computed using Magma) and their sum are as follows1 + / + / + / + / + / + w + w ′ = , where w , w ′ ∈ Q ( √
5) are the roots of3025 x − x + = . (The corresponding character dimensions χ (1) are 1 , , , , , , , w = x yxy ,it did indeed seem that s χ ( w ) χ (1) ∈ Z for all χ . It turns out not di ffi cult to show why. Consider moregenerally w = x − m yx n y − (take m = − , n = x by xy to obtain a conjugate of the word x yxy just mentioned). The resulting group ˜ Γ is called a Baumslag–Solitar group.Let V be an irreducible representation of G over C with character χ . For a fixed u ∈ G let U : = | G | X y ∈ G yuy − ∈ End( V ) . It is easy to check that U preserves the G action on V . Hence by Schur’s lemma it must be of the form γ id V . Computing traces we find that γ = χ ( u ) /χ (1). Setting u = x n and computing the trace of x − m U wefinally find that χ (1) s χ ( w ) = h Ψ m χ, Ψ n χ i , where h α, β i : = / | G | P x ∈ G α ( x ) ¯ β ( x ) is the standard inner product of class functions and Ψ m ( χ )( x ) : = χ ( x m )is the m -th Adams operations on (virtual) characters. In particular, χ (1) s χ ( w ) is an integer for all χ .More directly, if n = yxy − = x m with x , y ∈ G as follows. Group thesolutions according to the conjugacy class C of x . This class must satisfy C m = C and contributes precisely | G | to the total since for fixed x ∈ C we have that y lies in a coset of the centralizer of x . Hence in this case1 | G | Γ , G ) = { C m = C } . For example, for G = SL ( F p ) it is not hard to verify that when m is even this number equals1 + δ p ( m ) + X ε ,ε = ± (gcd( p + ε , m + ε ) − , where δ p ( m ) = m is a square modulo p and is zero otherwise. It is a consequence of the exponential formula in combinatorics that for a finitely generated group ˜ Γ wehave F ( x ) : = X n ≥ Γ , S n ) x n n ! = exp X n ≥ u n ( ˜ Γ ) x n n , where u n ( ˜ Γ ) denotes the number of subgroups of ˜ Γ of index n (see for example [2],[5]). By Corollary 3.3if ˜ Γ has infinite abelianization then the series on the left hand side has integer coe ffi cients and hence maybe written as an infinite product F ( x ) = Y n ≥ (1 − x n ) − v n (˜ Γ ) , for certain integers v n ( ˜ Γ ). Comparing the two expressions we find that u n = X d | n d v d (5.0.5)and by M¨obius inversion v n = n X d | n µ (cid:18) nd (cid:19) u d . (5.0.6)As before we can write ˜ Γ = Γ ⋊ σ Z and by Corollary 3.2 we get F ( x ) = X n ≥ σ ( Γ , S n ) / S n ) x n . We can interpret the n -th coe ffi cient of this series as the number of isomorphism classes of σ -equivariantactions of Γ on a set of n elements. It follows that the exponents v n ( ˜ Γ ) count the number of such actionswhich are indecomposable (i.e., transitive). In particular, v n ( ˜ Γ ) is a non-negative integer.If σ is trivial, equivalently if ˜ Γ = Γ × Z is a direct product, then isomorphism classes of transitive actionsof Γ on n objects corresponds to conjugacy classes of subgroups of Γ of index n . (This interpretation of v n ( ˜ Γ ) appears as exercise 5.13 (c) in [5] with a di ff erent suggested proof.)From (5.0.5), we obtain the following. Proposition 5.1.
Let ˜ Γ be a finitely generated group with infinite abelianization and let u n : = u n ( ˜ Γ ) be itsnumber of subgroups of index n. Then for every prime number p we haveu p k + ≡ u p k mod p k + . Here is a short table of the numbers u n in the case of ˜ Γ = π ( Σ g ), the fundamental group of a genus g Riemann surface, g \ n v n g \ n References [1] T. Hausel, E. Letellier and F. Rodriguez Villegas,
Arithmetic harmonic analysis on character and quiver varieties (to appear in Duke Math. J.) 4[2] A. D. Mednykh,
On the solution of the Hurwitz problem on the number of nonequivalent coverings over a compactRiemann surface (Russian) Dokl. Akad. Nauk SSSR (1981), 537–542. 5[3] J.-P. Serre,
Topics in Galois theory
Research Notes in Mathematics, A K Peters, Ltd., Wellesley, MA, 2008 1, 4[4] L. Solomon,
The solution of equations in groups
Arch. Math. (1969) 241–247 1, 3.6[5] R. Stanley, Enumerative combinatorics. Vol. 2
Cambridge Studies in Advanced Mathematics,62