We determine the structure of the first resonance variety of the cohomology
ring of the group of automorphisms of a finitely generated free group which act
by conjugation on a given basis.Comment: 7 page

Cohen, Daniel C.

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Daniel C. Cohen

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Resonance of basis-conjugating automorphism groups

We determine the structure of the first resonance variety of the cohomology ring of the group of automorphisms of a finitely generated free group which act by conjugation on a given basis. © 2009 American Mathematical Society

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Louisiana State University LSU Scholarly Repository Faculty Publications Department of Mathematics 9-1-2009 Resonance of basis-conjugating automorphism groups Daniel C. Cohen Louisiana State University Follow this and additional works at: https://repository.lsu.edu/mathematics_pubs Recommended Citation Cohen, D. (2009). Resonance of basis-conjugating automorphism groups. Proceedings of the American Mathematical Society, 137 (9), 2835-2841. https://doi.org/10.1090/S0002-9939-09-09858-X This Article is brought to you for free and open access by the Department of Mathematics at LSU Scholarly Repository. It has been accepted for inclusion in Faculty Publications by an authorized administrator of LSU Scholarly Repository. For more information, please contact ir@lsu.edu. PROCEEDINGS OF THEAMERICAN MATHEMATICAL SOCIETYVolume 137, Number 9, September 2009, Pages 2835–2841S 0002-9939(09)09858-XArticle electronically published on February 16, 2009RESONANCE OF BASIS-CONJUGATINGAUTOMORPHISM GROUPSDANIEL C. COHEN(Communicated by Daniel Ruberman)Abstract. We determine the structure of the first resonance variety of thecohomology ring of the group of automorphisms of a finitely generated freegroup which act by conjugation on a given basis.1. Resonance of PΣnLet Fn be the free group generated by x1, . . . , xn. The basis-conjugating auto-morphism group, or pure symmetric automorphism group, is the group PΣn of allautomorphisms of Fn which send each generator xi to a conjugate of itself. Resultsof Dahm [4] and Goldsmith [7] imply that this group may also be realized as the“group of loops”, the group of motions of a collection of n unknotted, unlinked ori-ented circles in 3-space, where each circle returns to its original position. McCool[9] found the following presentation for the basis-conjugating automorphism group:(1.1) PΣn = 〈βi,j , 1 ≤ i = j ≤ n | [βi,j , βk,l], [βi,k, βj,k], [βi,j , (βi,k · βj,k)]〉,where [u, v] = uvu−1v−1 denotes the commutator, the indices in the relations aredistinct, and the generators βi,j are the automorphisms of Fn defined byβi,j(xk) ={xk if k = j,x−1j xixj if k = i.The purpose of this paper is to determine the structure of the first resonance varietyof the cohomology ring of this group.Let A =⊕k=0 Ak be a finite-dimensional, graded, connected algebra over analgebraically closed field k of characteristic 0. Since a · a = 0 for each a ∈ A1,multiplication by a defines a cochain complex (A, δa):A0δa−−−−−→ A1 δa−−−−−→ A2 δa−−−−−→ · · · · · · δa−−−−−→ A,where δa(x) = ax. The resonance varieties of A are the jumping loci for the co-homology of these complexes: Rjd(A) ={a ∈ A1 | dimk Hj(A, δa) ≥ d}. As shownby Falk [6], these algebraic subvarieties of A1 are isomorphism-type invariants ofthe algebra A. They have been the subject of considerable recent interest in thecontext of hyperplane arrangements and related areas; see, for instance, Dimca,Received by the editors November 17, 2008.2000 Mathematics Subject Classification. Primary 20F28; Secondary 14M12, 20J06.The author was partially supported by National Security Agency grant No. H98230-05-1-0055.c©2009 American Mathematical SocietyReverts to public domain 28 years from publication28352836 DANIEL C. COHENPapadima, and Suciu [5], Yuzvinsky [14], and the references therein. We will focuson the first resonance variety R1(A) ={a ∈ A1 | H1(A, δa) = 0}.Since the relations in the presentation (1.1) of PΣn are all commutators, thefirst homology group H1(PΣn; k) is a vector space of dimension n(n−1) with basis{[βp,q] | 1 ≤ p = q ≤ }. Let {ep,q | 1 ≤ p = q ≤ } be the dual basis of H1(PΣn; k).Denote the first resonance variety of A = H∗(PΣn; k) by R1(PΣn, k).Theorem 1.1. The first resonance variety of the cohomology ring H∗(PΣn; k) ofthe basis-conjugating automorphism group isR1(PΣn, k) =⋃1≤i<j≤nCi,j ∪⋃1≤i<j<k≤nCi,j,k,where Ci,j = span {ei,j , ej,i} and Ci,j,k = span {ej,i − ek,i, ei,j − ek,j , ei,k − ej,k}.This result reveals an interesting relationship between the resonance variety andthe Bieri-Neumann-Strebel (BNS) invariant of the basis-conjugating automorphismgroup. For a finitely generated group G, let C be the Cayley graph correspondingto a finite generating set. Given an additive character χ : G → R, let C+(χ) bethe full subgraph of C on the vertex set {g ∈ G | χ(g) ≥ 0}. Then the (first) BNSinvariant of G is the conical subset Σ(G) of Hom(G, R) = H1(G; R) defined byΣ(G) = {χ ∈ Hom(G, R) \ {0} | C+(χ) is connected} .This invariant of G (which is independent of the choice of generating set) maybe used to determine which subgroups above the commutator subgroup [G, G] arefinitely generated; see [1].The BNS invariant of the group G = PΣn was determined by Orlandi-Korner[11]. Combining her result with the above theorem yields the following:Theorem 1.2. The Bieri-Neumann-Strebel invariant of the basis-conjugating au-tomorphism group is given byΣ(PΣn) = H1(PΣn; R) \ R1(PΣn, R).This relationship between the resonance variety and the BNS invariant is knownto hold for certain other groups, including right-angled Artin groups; see Meierand VanWyk [10] and Papadima and Suciu [12]. More recently, Papadima andSuciu have shown that the containment Σ(G) ⊆ H1(G; R) \ R1(G, R) holds for anarbitrary 1-formal group G, where, as above, R1(G; R) denotes the first resonancevariety of H∗(G; R). See [13] for a detailed investigation of the relationship betweenBNS invariants and (co)homology jumping loci in a number of contexts.2. Proof of Theorem 1.1In this section, we recall the structure of the cohomology ring of the basis-conjugating automorphism group and use it to prove Theorem 1.1. The cohomologyof PΣn was computed by Jensen, McCammond, and Meier [8], resolving positivelya conjecture of Brownstein and Lee [2].Theorem 2.1 ([8]). Let EZ denote the exterior algebra over Z generated by degreeone elements ep,q, 1 ≤ p = q ≤ n, and let IZ denote the two-sided ideal in EZgenerated by(2.1)ηi,j = ei,jej,i, 1 ≤ i < j ≤ n;τki,j = (ek,i − ej,i)(ek,j − ei,j), 1 ≤ k ≤ n, 1 ≤ i < j ≤ n, k /∈ {i, j} .RESONANCE OF BASIS-CONJUGATING AUTOMORPHISM GROUPS 2837Then the integral cohomology algebra of the basis-conjugating automorphism groupPΣn is isomorphic to the quotient of EZ by IZ, H∗(PΣn; Z) ∼= EZ/IZ.Remark 2.2. The above presentation of the cohomology ring H∗(PΣn; Z) differsslightly from that given in [2, 8] but is easily seen to be equivalent. For instance,the relation in H∗(PΣn; Z) arising from the generator τki,j of the ideal I may beobtained by constructing an appropriate linear combination of the relations labeled2 and 3 in [8, Thm. 6.7].Let k be a field of characteristic zero. From Theorem 2.1 and the UniversalCoefficient Theorem, the cohomology algebra H∗(PΣn; k) is isomorphic to Ek/Ik,where Ek is the exterior algebra over k generated by ep,q , 1 ≤ p = q ≤ n, and Ik isthe ideal in Ek generated by the elements ηi,j and τki,j above.Recall from Section 1 that the first resonance variety of A = H∗(PΣn; k) isR1(PΣn, k) = {a ∈ A1 | H1(A, δa) = 0}.Observe that A1 = H1(PΣn; k) is a vector space of dimension N = n(n − 1) overk. Elements of A1 are of the form a =∑p=q ap,qep,q, where ap,q ∈ k. Recall that(2.2) Ci,j = span {ei,j , ej,i} = {a ∈ A1 | ap,q = 0 if {p, q} = {i, j}}for 1 ≤ i < j ≤ n and that(2.3)Ci,j,k = span {ej,i − ek,i, ei,j − ek,j , ei,k − ej,k}={a ∈ A1∣∣∣∣∣ aj,i + ak,i = 0, ai,j + ak,j = 0, ai,k + aj,k = 0,ap,q = 0 if {p, q} ⊂ {i, j, k}}for 1 ≤ i < j < k ≤ n. Note that these are linear subspaces of A1 ∼= kN of dimen-sions 2 and 3 respectively. To prove Theorem 1.1, we must show that R1(PΣn, k)is equal to the union of these linear subspaces.Proof of Theorem 1.1. In the case n = 2, the group PΣn = 〈β1,2, β2,1〉 ∼= F2 is afree group, and the theorem asserts that R1(PΣ2, k) = C1,2 = H1(PΣ2; k), whichis clear. So assume that n ≥ 3.Write R = R1(PΣn, k) and C =⋃1≤i<j≤n Ci,j ∪⋃1≤i<j<k≤n Ci,j,k. Observethat 0 ∈ C and 0 ∈ R. So it is enough to show that R \ {0} = C \ {0}.Write E = Ek and I = Ik. For a ∈ A1 = E1, we have a short exact sequence ofchain complexes 0 −→ (I, δa)ι−−→ (E, δa)p−−→ (A, δa) −→ 0:I2ι2δa  I3ι3 . . .E0p0δa  E1p1δa  E2p2δa  E3p3 . . .A0δa  A1δa  A2δa  A3  . . .where ι : I → E is the inclusion, p : E → A the projection, and δa(x) = ax. Notethat since I is generated in degree two, the maps p0 : E0 → A0 and p1 : E1 → A1are identity maps. If a = 0, then the complex (E, δa) is acyclic. Consequently, thecorresponding long exact cohomology sequence yieldsH1(A, δa) ∼= H2(I, δa) = ker(δa : I2 → I3) = ker(ι3 ◦ δa : I2 → E3).2838 DANIEL C. COHENThus, a ∈ R \ {0} if and only if the map ψa := ι3 ◦ δa fails to inject.Since the elements ηi,j and τki,j recorded in (2.1) generate the ideal I and are ofdegree two (and are linearly independent in E2), these elements form a basis for I2.We record the images of these basis elements under the map ψa. For 1 ≤ i < j ≤ n,(2.4) ψa(ηi,j) =∑{p,q}={i,j}ap,qep,qηi,j =∑{p,q}={i,j}ap,qep,qei,jej,i.For 1 ≤ k ≤ n, 1 ≤ i < j ≤ n, k /∈ {i, j},(2.5)ψa(τki,j) = (aj,i + ak,i)ej,iek,i(ek,j − ei,j) − (ai,j + ak,j)ei,jek,j(ek,i − ej,i)+ (ai,kei,k + aj,kej,k)τki,j +∑{p,q}⊂{i,j,k}ap,qep,qτki,j .These calculations immediately yield the containment C \ {0} ⊆ R \ {0}. Ifa ∈ Ci,j , then ap,q = 0 for {p, q} = {i, j}. For such an a, we have ψa(ηi,j) = 0by (2.4), so Ci,j ⊂ R. If 1 ≤ i < j < k ≤ n and a ∈ Ci,j,k, then aj,i + ak,i = 0,ai,j + ak,j = 0, ai,k + aj,k = 0, and ap,q = 0 for {p, q} ⊂ {i, j, k}. In this instance,(2.5) may be used to check that aj,iτki,j − ai,kτ ij,k and ai,jτki,j − ai,kτji,k are elementsof ker(ψa). If a ∈ Ci,j,k is nonzero, at least one of ai,j , ai,k, aj,i must be nonzero.Consequently, ψa has nontrivial kernel, and Ci,j,k ⊂ R.Establishing the reverse containment, R \ {0} ⊆ C \ {0}, is more involved. Wewill show that a /∈ C implies that a /∈ R. If a /∈ C, then a = 0. So assumewithout loss that a2,1 = 0. Since a /∈ C1,2 ⊂ C, we must also have ap,q = 0 forsome {p, q} = {1, 2}. We will consider several cases depending on the relationshipbetween the sets {1, 2} and {p, q}.Case 1. {1, 2} ∩ {p, q} = ∅Assume first that {1, 2} and {p, q} are disjoint. Note that n ≥ 4 in this instance.Permuting indices if need be, we may assume that a ∈ H1(PΣn; k) satisfies a2,1 = 0and a3,4 = 0. We will show that this assumption implies that the map ψa : I2 → E3injects; hence a /∈ R. Specifically, we will exhibit a subspace V ⊂ E3 and aprojection π : E3  V so that the composition π ◦ ψa : I2 → V is an isomorphism.Let V be the union of the sets{e1,2e2,1e3,4} ∪ {e2,1ei,jej,i | 1 ≤ i < j ≤ n, {i, j} = {1, 2}} ,{e3,4e1,2ek,1, e3,4e2,1e1,k, e3,4e1,2e1,k | 3 ≤ k ≤ n} ,{e2,1ek,iek,j , e2,1ej,iej,k, e2,1ei,kek,j | i ≤ 2 < j < k ≤ n or 3 ≤ i < j < k ≤ n} .There is a bijection between V and the set of generators of I2 given byη1,2 ↔ e1,2e2,1e3,4, ηi,j ↔ e2,1ei,jej,i ({i, j} = {1, 2}),τk1,2 ↔ e3,4e1,2ek,1, τ21,k ↔ e3,4e2,1e1,k, τ12,k ↔ e3,4e1,2e1,k (3 ≤ k ≤ n),τki,j ↔ e2,1ek,iek,j , τji,k ↔ e2,1ej,iej,k, τ ij,k ↔ e2,1ei,kek,j (i ≤ 2 < j < k ≤ n).In particular, the monomials in V are distinct. Hence, V is a linearly independentset in E3 of cardinality |V| = dimk I2 =(n2)(n − 1). Let V = spanV ⊂ E3.Define π : E3  V on basis elements byπ(ea,bec,dep,q) ={ea,bec,dep,q if ea,bec,dep,q ∈ V ,0 if ea,bec,dep,q /∈ V .RESONANCE OF BASIS-CONJUGATING AUTOMORPHISM GROUPS 2839Then, a calculation using (2.4) and (2.5) reveals that π ◦ ψa : I2 → V is an iso-morphism. For instance, ordering the bases of I2 and V appropriately, one cancheck that the matrix M of π ◦ ψa has determinant detM = ym2,12,1 ym3,43,4 = 0, wherem3,4 = 3n − 5 and m2,1 =(n2)(n − 1) − m3,4. Thus, if a /∈ C satisfies a2,1 = 0 andap,q = 0 for some p, q with {1, 2} ∩ {p, q} = ∅, then a /∈ R.Case 2. {1, 2} ∩ {p, q} = ∅Now assume that a /∈ C, a2,1 = 0, and ar,s = 0 for all r, s with {1, 2} ∩ {r, s} = ∅.Since a /∈ C1,2 ⊂ C, we must have ap,q = 0 for some p, q with |{p, q} ∩ {1, 2}| = 1.Permuting indices if necessary, we may assume that 3 ∈ {p, q}.In the case n = 3, since a /∈ C1,2,3 ⊂ C, one of the sums a2,1 + a3,1, a1,2 + a3,2,a1,3+a2,3 must be nonzero; see (2.3). In this instance, ordering bases appropriately,the map ψa : I2 → E3 has matrix(2.6) M3 =⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝a3,2 0 0 0 −a1,2 − a3,2 0a3,1 0 0 0 −a2,1 − a3,1 0a2,3 0 0 a1,2 −a2,3 a2,1−a1,3 0 0 a1,2 a1,3 a2,10 a3,2 0 −a3,2 a1,3 −a3,10 −a2,3 0 a1,3 + a2,3 0 00 −a2,1 0 a2,1 + a3,1 0 00 a1,2 0 −a1,2 −a1,3 a3,10 0 −a3,1 a3,2 a2,3 a3,10 0 a2,1 a3,2 a2,3 −a2,10 0 a1,3 0 0 −a1,3 − a2,30 0 a1,2 0 0 −a1,2 − a3,20 0 0 a3,2 −a1,3 −a2,10 0 0 a2,1 + a3,1 0 00 0 0 a1,3 + a2,3 0 00 0 0 a1,2 −a2,3 −a3,10 0 0 0 a1,2 + a3,2 00 0 0 0 a2,1 + a3,1 00 0 0 0 0 a1,2 + a3,20 0 0 0 0 a1,3 + a2,3⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.Using the assumptions on a2,1, the sums a2,1 + a3,1, a1,2 + a3,2, a1,3 + a2,3, andap,q ∈ {a1,3, a2,3, a3,1, a3,2}, one can check that the matrix M3 has maximal rank.Hence, if a /∈ C, then ψa : I2 → E3 injects in the case n = 3.For general n, the assumption that a /∈ C1,2,3 implies that the set{a2,1 + a3,1, a1,2 + a3,2, a1,3 + a2,3} ∪ {ar,s | {r, s} ⊂ {1, 2, 3}}contains a nonzero element. Recall that, by Case 1, we may assume that ar,s = 0for all r, s with {1, 2} ∩ {r, s} = ∅. If ar,s = 0 for all {r, s} ⊂ {1, 2, 3}, let W be thesubspace of E3 spanned by the union of the sets{ei1,j1ei2,j2ei3,j3 | 1 ≤ ik, jk ≤ 3, ik = jk} ,{e1,2e2,1ep,q} ∪ {e2,1ei,jej,i | 1 ≤ i < j ≤ n, {i, j} = {1, 2}} ,{e2,1ek,iek,j , e2,1ej,iej,k, e2,1ei,kek,j | i ≤ 2 < j < k ≤ n or 3 ≤ i < j < k ≤ n} ,⎧⎨⎩ep,qek,1ek,2, ep,qek,1e1,2, ep,qe2,1ek,2,ep,qe2,1e2,k, ep,qe2,1e1,k, ep,qek,1e2,k,ep,qe1,2e1,k, ep,qe1,2e2,k, ep,qek,2e1,k∣∣∣∣∣ 3 ≤ k ≤ n⎫⎬⎭ .2840 DANIEL C. COHENDefine π : E3  W on basis elements as before. Ordering bases appropriately, onecan use (2.4) and (2.5) to find a submatrix M of the matrix of π ◦ ψa : I2 → W ofthe formM =(U ∗0 M3),where M3 is given by (2.6) and U is upper triangular, with diagonal entries a2,1 = 0and ap,q = 0. (The choice of U depends on which ap,q ∈ {a1,3, a2,3, a3,1, a3,2} isnonzero.) Hence, the matrix M has maximal rank. It follows that ψa : I2 → E3injects.Finally, consider the case where a /∈ C1,2,3 ⊂ C, a2,1 = 0, ap,q = 0 for someap,q ∈ {a1,3, a2,3, a3,1, a3,2}, and ar,s = 0 for some {r, s} ⊂ {1, 2, 3}. Since we mayassume by Case 1 that ar,s = 0 if {1, 2} ∩ {r, s} = ∅, we have ar,s = 0 for somer, s with r ∈ {1, 2} and 4 ≤ s ≤ n. In this instance, let W be the subspace of E3spanned by the union of the sets{e1,2e2,1ep,q} ∪ {e2,1ei,jej,i | 1 ≤ i < j ≤ n, {i, j} = {1, 2}} ,{e2,1ek,iek,j , e2,1ej,iej,k, e2,1ei,kek,j | i ≤ 2 < j < k ≤ n or 3 ≤ i < j < k ≤ n} ,{er,se3,1e3,2, er,se1,3e3,2, er,se2,3e3,1} ,{ep,qek,1ek,2, ep,qe2,kek,1, ep,qe1,kek,2 | 4 ≤ k ≤ n} .Defining π : E3  W on basis elements as above, a calculation using (2.4) and(2.5) shows that π ◦ ψa : I2 → W is an isomorphism. For instance, ordering basesappropriately, one can check that the matrix M of π ◦ ψa has determinant detM =am2,12,1 amp,qp,q a3r,s, where mp,q = 3n − 8 and m2,1 =(n2)(n − 1) − mp,q. Hence, ψainjects in this final case.Thus, for any a /∈ C, the map ψa : I2 → E3 injects, and a /∈ R. This completesthe proof of Theorem 1.1. Remark 2.3. It follows from Theorem 2.1 that the integral cohomology groups ofPΣn are torsion free, with Betti numbers bk(PΣn) = rank Hk(PΣn; Z) given by thecoefficients of the Poincaré polynomial p(PΣn, t) =∑k≥0 bk(PΣn)·tk = (1+nt)n−1;see [8, §6]. Thus the cohomology groups cannot distinguish PΣn from a directproduct Fn × · · · × Fn of n − 1 free groups of rank n. For n = 2, this is to beexpected, since PΣ2 ∼= F2.For n ≥ 3, the groups PΣn and Fn × · · · × Fn are not isomorphic and are, infact, distinguished by their cohomology rings. By Theorem 1.1, the irreduciblecomponents of R1(PΣn, k) are two- and three-dimensional. On the other hand, theresults of [3] or [12] may be used to show that the irreducible components of thefirst resonance variety of H∗(Fn × · · · × Fn; k) are all n-dimensional.AcknowledgmentsWe thank Graham Denham and Michael Falk for useful conversations.References1. R. Bieri, W. D. Neumann, R. Strebel, A geometric invariant of discrete groups, Invent. Math.90 (1987), 451–477. MR09148462. A. Brownstein, R. Lee, Cohomology of the group of motions of n strings in 3-space, in: Map-ping class groups and moduli spaces of Riemann surfaces (Goettingen, 1991 / Seattle, WA,1991), pp. 51–61, Contemp. Math. 150, Amer. Math. Soc., Providence, RI, 1993. MR1234259RESONANCE OF BASIS-CONJUGATING AUTOMORPHISM GROUPS 28413. D. Cohen, A. Suciu, Characteristic varieties of arrangements, Math. Proc. Cambridge Phil.Soc. 127 (1999), 33–53. MR16925194. D. Dahm, A generalization of a braid theory, Ph.D. Thesis, Princeton Univ., 1962.5. A. Dimca, S. Papadima, A. Suciu, Topology and geometry of cohomology jumping loci, DukeMath. J., to appear. arXiv:0902.12506. M. Falk, Arrangements and cohomology, Ann. Combin. 1 (1997), 135–157. MR16296817. D. Goldsmith, The theory of motion groups, Michigan Math. J. 28 (1981), 3–17. MR06004118. C. Jensen, J. McCammond, J. Meier, The integral cohomology of the group of loops, Geom.Topol. 10 (2006), 759–784. MR22409059. J. McCool, On basis-conjugating automorphisms of free groups, Canad. J. Math. 38 (1986),1525–1529. MR087342110. J. Meier, L. VanWyk, The Bieri-Neumann-Strebel invariants for graph groups, Proc. LondonMath. Soc. (3) 71 (1995), 263–280. MR133746811. L. Orlandi-Korner, The Bieri-Neumann-Strebel invariant for basis-conjugating automor-phisms of free groups, Proc. Amer. Math. Soc. 128 (2000), 1257–1262. MR171288912. S. Papadima, A. Suciu, Algebraic invariants for right-angled Artin groups, Math. Ann. 334(2006), 533–555. MR220787413. S. Papadima, A. Suciu, Bieri-Neumann-Strebel-Renz invariants and homology jumping loci,preprint, 2008. arXiv:0812.266014. S. Yuzvinsky, Orlik-Solomon algebras in algebra and topology, Russian Math. Surveys 56(2001), 293–364. MR1859708Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana70803E-mail address: cohen@math.lsu.eduURL: http://www.math.lsu.edu/~cohen/

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Resonance of basis-conjugating automorphism groups

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