On the second homology group of the Torelli subgroup of Aut(F_n)
aa r X i v : . [ m a t h . G T ] D ec On the second homology group of the Torelli subgroup ofAut p F n q Matthew Day ˚ and Andrew Putman : Abstract
Let IA n be the Torelli subgroup of Aut p F n q . We give an explicit finite set of gener-ators for H p IA n q as a GL n p Z q -module. Corollaries include a version of surjective rep-resentation stability for H p IA n q , the vanishing of the GL n p Z q -coinvariants of H p IA n q ,and the vanishing of the second rational homology group of the level ℓ congruence sub-group of Aut p F n q . Our generating set is derived from a new group presentation for IA n which is infinite but which has a simple recursive form. The
Torelli subgroup of the automorphism group of a free group F n on n letters, denotedIA n , is the kernel of the action of Aut p F n q on F ab n – Z n . The group of automorphisms of Z n is GL n p Z q and the resulting map Aut p F n q Ñ GL n p Z q is easily seen to be surjective, sowe have a short exact sequence1 ÝÑ IA n ÝÑ Aut p F n q ÝÑ GL n p Z q ÝÑ . Though it has a large literature, the cohomology and combinatorial group theory of IA n remain quite mysterious. Magnus [28] proved that IA n is finitely generated, and thus thatH p IA n q has finite rank. Krsti´c–McCool [26] later showed that IA is not finitely presentable.This was improved by Bestvina–Bux–Margalit [4], who showed that H p IA q has infiniterank. However, for n ě n is finitely presentable orwhether or not H p IA n q has finite rank. Representation-theoretic finiteness.
It seems to be very difficult to determine whetheror not H p IA n q has finite rank, so it is natural to investigate weaker sorts of finitenessproperties. Since inner automorphisms act trivially on homology, the conjugation action ofAut p F n q on IA n induces an action of GL n p Z q on H k p IA n q . Church–Farb [12, Conjecture 6.7]conjectured that H k p IA n q is finitely generated as a GL n p Z q -module. In other words, theyconjectured that there exists a finite subset of H k p IA n q whose GL n p Z q -orbit spans H k p IA n q .Our first main theorem verifies their conjecture for k “ Theorem A (Generators for H p IA n q ) . For all n ě , there exists a finite subset of H p IA n q whose GL n p Z q -orbit spans H p IA n q . Each element of our finite subset corresponds to a map of a surface into a classifying spacefor IA n ; the genera of these surfaces range from 1 to 3. Table 1 below lists our finiteset of GL n p Z q -generators for H p IA n q . This table expresses these generators using specific“commutator relators” in IA n ; see below for how to translate these into elements of H p IA n q . ˚ Supported in part by NSF grant DMS-1206981 : Supported in part by NSF grant DMS-1255350 and the Alfred P. Sloan Foundation emark . The special case n “ n . Surjective representation stability.
The generators for H p IA n q given in Theorem Aare explicit enough that they can be used to perform a number of interesting calculations.The first verifies part of a conjecture of Church–Farb that asserts that the homology groupsof IA n are “representation stable”. We begin with some background. An increasing sequence G Ă G Ă G Ă ¨ ¨ ¨ of groups is homologically stable if for all k ě
1, the k th homology group of G n is independentof n for n "
0. Many sequences of groups are homologically stable; see [22] for a bibliography.In particular, Hatcher–Vogtmann [21] proved this for Aut p F n q . However, it is known thatIA n is not homologically stable; indeed, even H p IA n q does not stabilize (see below).Church–Farb [12] introduced a new form of homological stability for groups like IA n whosehomology groups possess natural group actions. For IA n , they conjectured that for all k ě n k ě n ě n k . • (Injective stability) The map H k p IA n q Ñ H k p IA n ` q is injective. • (Surjective representation stability) The map H k p IA n q Ñ H k p IA n ` q is surjective “upto the action of GL n ` p Z q ”; more precisely, the GL n ` p Z q -orbit of its image spansH k p IA n ` q . Remark . In fact, they made this conjecture in [11] for the Torelli subgroup of themapping class group; however, they have informed us that they also conjecture it for IA n .Our generators for H p IA n q are “the same in each dimension” starting at n “
6, so we areable to derive the following special case of Church–Farb’s conjecture.
Theorem B (Surjective representation stability for H p IA n q ) . For n ě , the GL n ` p Z q -orbit of the image of the natural map H p IA n q Ñ H p IA n ` q spans H p IA n ` q .Remark . Boldsen–Dollerup [5] proved a theorem similar to Theorem B for the rational second homology group of the Torelli subgroup of the mapping class group. Their proof isdifferent from ours; in particular, they were not able to prove an analogue of Theorem A.It seems hard to use their techniques to prove Theorem B. Similarly, our proof uses specialproperties of IA n and does not work for the Torelli subgroup of the mapping class group. Coinvariants.
Our tools do not allow us to easily distinguish different homology classes;indeed, for all we know our generators for H p IA n q might be redundant. This prevents usfrom proving injective stability for H p IA n q . However, we still can prove some interestingvanishing results. If G is a group and M is a G -module, then the coinvariants of G actingon M , denoted M G , are the largest quotient of M on which G acts trivially. More precisely, M G “ M { K with K “ x m ´ g ¨ m | g P G , m P M y . We then have the following. Theorem C (Vanishing coinvariants) . For n ě , we have p H p IA n qq GL n p Z q “ .Remark . In [12, Conjecture 6.5], Church–Farb conjectured that the GL n p Z q -invariantsin H k p IA n ; Q q are 0. For k “
1, this follows from the known computation of H p IA n ; Q q ;see below. Theorem C implies that this also holds for k “ inear congruence subgroups. For ℓ ě
2, the level ℓ congruence subgroup of Aut p F n q ,denoted Aut p F n , ℓ q , is the kernel of the natural map Aut p F n q Ñ GL n p Z { ℓ q ; one shouldthink of it as a “mod- ℓ ” version of IA n . It is natural to conjecture that for all k ě
1, thereexists some n k ě k p Aut p F n , ℓ q ; Q q – H k p Aut p F n q ; Q q for n ě n k ; an analogoustheorem for congruence subgroups of GL n p Z q is due to Borel [7]. Galatius [20] proved thatH k p Aut p F n q ; Q q “ n "
0, so this conjecture really asserts that H k p Aut p F n , ℓ q ; Q q “ n "
0. The case k “ p F n , ℓ q for n ě p Aut p F n , ℓ q ; Q q “ n ě
3. Using Theorem A, we will prove the case k “ Theorem D (Second homology of congruence subgroups) . For ℓ ě and n ě , we have H p Aut p F n , ℓ q ; Q q “ . The key to our proof is that Theorem A allows us to show that the image of H p IA n ; Q q inH p Aut p F n , ℓ q ; Q q vanishes; this allows us to derive Theorem D using standard techniques. Remark . The second author proved an analogue of Theorem D for congruence subgroupsof the mapping class group in [32]. The techniques in [32] are different from those in thepresent paper and it seems difficult to prove Theorem D via those techniques.
Basic elements of Torelli.
We now wish to describe our generating set for H p IA n q .This requires introducing some basic elements of IA n . Let t x , . . . , x n u be a free basis for F n . We then make the following definitions. • For distinct 1 ď i, j ď n , let C x i ,x j P IA n be defined via the formulasC x i ,x j p x i q “ x j x i x ´ j and C x i ,x j p x ℓ q “ x ℓ if ℓ ‰ i. • For α, β, γ
P t˘ u and distinct 1 ď i, j, k ď n , let M x αi , r x βj ,x γk s P IA n be defined via theformulas M x αi , r x βj ,x γk s p x αi q “ r x βj , x γk s x αi and M x αi , r x βj ,x γk s p x ℓ q “ x ℓ if ℓ ‰ i. Observe that by definition M x ´ i , r x βj ,x γk s p x ´ i q “ r x βj , x γk s x ´ i and M x ´ i , r x βj ,x γk s p x i q “ x i r x βj , x γk s ´ . We call C x i ,x j a conjugation move and M x αi , r x βj ,x γk s a commutator transvection . Surfaces in a classifying space: our generators. A commutator relator in IA n is aformula of the form r a , b s ¨ ¨ ¨ r a g , b g s “ a i , b i P IA n . Given such a commutatorrelator r , let Σ g be a genus g surface. There is a continuous map ζ : Σ g Ñ K p IA n , q that takes the standard basis for π p Σ g q to a , b , . . . , a g , b g P IA n . We obtain an element h r “ ζ ˚ pr Σ g sq P H p IA n q . With this notation, the generators for H p IA n q given by TheoremA are the elements h r where r is one of the relators in Table 1.31. r C x a ,x b , C x c ,x d s “
1, possibly with b “ d .H2. r M x αa , r x βb ,x γc s , M x δd , r x ǫe ,x ζf s s “
1, possibly with t b, c u X t e, f u ‰ ∅ or with x αa “ x ´ δd ,as long as x αa ‰ x δd , a R t e, f u and d R t b, c u .H3. r C x a ,x b , M x γc , r x δd ,x ǫe s s “
1, possibly with b P t d, e u , if c R t a, b u and a R t c, d, e u .H4. r C βx c ,x b C βx a ,x b , C αx c ,x a s “ r C ´ γx a ,x c , C ´ δx a ,x d sr C ´ βx a ,x b , M x βb , r x γc ,x δd s s “ r M x αa , r x βb ,x γc s , M x δd , r x αa ,x ǫe s sr M x δd , r x αa ,x ǫe s , M x δd , r x γc ,x βb s sr M x δd , r x γc ,x βb s , C ´ ǫx d ,x e s “
1, possiblywith b “ e or c “ e .H7. r M x γc , r x αa ,x δd s , C βx a ,x b sr C ´ δx c ,x d , M x γc , r x αa ,x βb s sr M x γc , r x αa ,x βb s , M x γc , r x αa ,x δd s s “
1, possibly with b “ d .H8. r M x αa , r x βb ,x γc s , C δx a ,x d C δx b ,x d C δx c ,x d s “ r C γx a ,x c C γx b ,x c , C βx a ,x b C βx c ,x b sr M x αa , r x βb ,x γc s , C αx b ,x a C αx c ,x a s “ Table 1:
The set of commutator relators whose associated elements of H p IA n q generate it as a GL n p Z q -module. Distinct letters represent distinct indices unless stated otherwise. The Johnson homomorphism.
To motivate our proof of Theorem A, we must firstrecall the computation of H p IA n q , which is due independently to Farb [19], Kawazumi[25], and Cohen–Pakianathan [14]. The basic tool is the Johnson homomorphism, whichwas introduced by Johnson [24] in the context of the Torelli subgroup of the mapping classgroup (though it also appears in earlier work of Andreadakis [1]). See [34] for a survey ofthe IA n -version of it. The Johnson homomorphism is a homomorphism τ : IA n Ñ Hom p Z n , Ź Z n q that arises from studying the action of IA n on the second nilpotent truncation of F n . It canbe defined as follows. For z P F n , let r z s P Z n be the associated element of the abelianizationof F n . Consider f P IA n . For x P F n , we have f p x q ¨ x ´ P r F n , F n s . There is a naturalsurjection ρ : r F n , F n s Ñ Ź Z n satisfying ρ pr a, b sq “ r a s^r b s ; the kernel of ρ is r F n , r F n , F n ss .We can then define a map ˜ τ f : F n Ñ Ź Z n via the formula ˜ τ f p x q “ ρ p f p x q ¨ x ´ q . One cancheck that ˜ τ f is a homomorphism. It factors through a homomorphism τ f : Z n Ñ Ź Z n .We can then define τ : IA n Ñ Hom p Z n , Ź Z n q via the formula τ p f q “ τ f . One can checkthat τ is a homomorphism. Generators and their images.
Define S MA p n q “ t C x i ,x j | ď i, j ď n distinct u Y t M x i , r x j ,x k s | ď i, j, k ď n distinct, j ă k u . Magnus [28] proved that IA n is generated by S MA p n q ; see [16] and [4] for modern proofs.For distinct 1 ď i, j ď n , the image τ p C x i ,x j q P Hom p Z n , Ź Z n q is the homomorphismdefined via the formulas r x i s ÞÑ r x j s ^ r x i s and r x ℓ s ÞÑ ℓ ‰ i. ď i, j, k ď n with j ă k , the image τ p M x i , r x j ,x k s q P Hom p Z n , Ź Z n q is the homomorphism defined via the formulas r x i s ÞÑ r x j s ^ r x k s and r x ℓ s ÞÑ ℓ ‰ i. The key observation is that these form a basis for Hom p Z n , Ź Z n q . The abelianization.
Let F p S MA p n qq be the free group on S MA p n q and let R MA p n q Ă F p S MA p n qq be a set of relations for IA n , so IA n “ x S MA p n q | R MA p n qy . Since τ takes S MA p n q bijectively to a basis for the free abelian group Hom p Z n , Ź Z n q , we must have R MA p n q Ăr F p S MA p n qq , F p S MA p n qqs . This immediately implies that H p IA n q – Hom p Z n , Ź Z n q . Hopf ’s formula.
But even more is true. Recall that Hopf’s formula [10] says that if G is a group with a presentation G “ x S | R y , thenH p G q – xx R yy X r F p S q , F p S qsr F p S q , xx R yys ;here xx R yy is the normal closure of R . The intersection in the numerator of this is usuallyhard to calculate, so Hopf’s formula is not often useful for computation. However, by whatwe have said it simplifies for IA n toH p IA n q – xx R MA p n qyyr F p S MA p n q , xx R MA p n qyys . (1)This isomorphism is very concrete: an element r P xx R MA p n qyy is a commutator relator, andthe associated element of H p IA n q is the homology class h r discussed above. Summary and trouble.
For r P xx R MA p n qyy and z P F p S MA p n qq , the element zrz ´ r ´ lies in the denominator of (1), i.e. r F p S MA p n qq , xx R MA p n qyys . Hence h zrz ´ “ h r . It followsthat H p IA n q is generated by the set t h r | r P R MA p n qu . In other words, to calculate gener-ators for H p IA n q , it is enough to find a presentation for IA n with S MA p n q as its generatingset. However, this seems like a difficult problem (especially if, as we suspect, IA n is notfinitely presentable). Moreover, the GL n p Z q -action on H p IA n q has not yet appeared. L-presentations.
To incorporate the GL n p Z q -action on H p IA n q into our presentationfor IA n , we use the notion of an L-presentation, which was introduced by Bartholdi [3] (weuse a slight simplification of his definition). An L-presentation for a group G is a triple x S | R | E y , where S and R and E are as follows. • S is a generating set for G . • R Ă F p S q is a set consisting of relations for G (not necessarily complete). • E is a subset of End p F p S qq .This data must satisfy the following condition. Let M Ă End p F p S qq be the monoid gener-ated by E . Define R “ t f p r q | f P M , r P R u . Then we require that G “ x S | R y . Eachelement of E descends to an element of End p G q ; we call the resulting subset r E Ă End p G q induced endomorphisms of our L-presentation. We say that our L-presentation is finite if the sets S and R and E are all finite.In our examples, the induced endomorphisms of our L-presentations will actually be auto-morphisms. Thus in the context of this paper one should think of an L-presentation as agroup presentation incorporating certain symmetries of a group. Here is an easy example. Example.
Let S “ t z i | i P Z { p u and R “ t z u . Let ψ : F p S q Ñ F p S q be the homomor-phism defined via the formula ψ p z i q “ z i ` . Then x S | R | t ψ uy is an L-presentation forthe free product of p copies of Z { A finite L-presentation for Torelli.
The conjugation action of Aut p F n q on IA n givesan injection Aut p F n q ã Ñ Aut p IA n q . If we could somehow construct a finite L-presentation x S MA p n q | R p n q | E MA p n qy for IA n whose set of induced endomorphisms generatedAut p F n q Ă Aut p IA n q Ă End p IA n q , then Theorem A would immediately follow. Indeed, since the GL n p Z q -action on H p IA n q isinduced by the conjugation action of Aut p F n q on IA n , it would follow that the GL n p Z q -orbitof the set t h r | r P R p n qu Ă H p IA n q spanned H p IA n q .Although we find the idea in the previous paragraph illuminating, we do not follow itstrictly. To make our L-presentation for IA n easier to comprehend, we will use the followinggenerating set, which is larger than S MA : S IA p n q “ t C x i ,x j | ď i, j ď n distinct uY t M x αi , r x βj ,x γk s | ď i, j, k ď n distinct, α, β, γ P t˘ uu . This has the advantage of making our relations and rewriting rules shorter, and making theirmeaning easier to understand. It has the disadvantage of making the proof of Theorem Aless direct. Our theorem giving an L-presentation for IA n is as follows. Theorem E (Finite L-presentation for Torelli.) . For all n ě , there exists a finite L-presentation IA n “ x S IA p n q | R p n q | E IA p n qy whose set of induced endomorphisms gener-ates Aut p F n q Ă Aut p IA n q Ă End p IA n q . We note that our presentation is not a presentation in which all relators are commutators.The formulas for the R p n q and E IA p n q in our finite L-presentation are a little complicated,so we postpone them until §
2. The formulas in that section make it clear that R IA p n q does not lie in r F p S IA p n qq , F p S IA p n qqs . Therefore we cannot prove Theorem A simply byinterepreting the relators as homology classes. We must do something more complicated todeduce that theorem from our presentation. Remark . The relations in Table 1 are not sufficient for our L-presentation. Indeed,they all lie in the commutator subgroup, but the generators S IA p n q do not map to linearlyindependent elements of the abelianization. Sketch of proof.
We close this introduction by briefly discussing how we prove TheoremE. In particular, we explain why it is easier to verify an L-presentation than a standard6resentation. We remark that our proof is inspired by a recent paper [9] of the secondauthor together with Brendle and Margalit which constructed generators for the kernel ofthe Burau representation evaluated at ´ x S IA p n q | R p n q | E IA p n qy for IA n asin Theorem E (we found the one that we use by first throwing in all the relations we couldthink of and then attempting the proof below; each time it failed it revealed a relation thatwe had missed). Let Q n be the group presented by the purported L-presentation. There isthus a surjection π : Q n Ñ IA n , and the goal of our proof will be to construct an inversemap φ : IA n Ñ Q n satisfying φ ˝ π “ id. This will involve several steps. Step 1.
We decompose IA n in terms of stabilizers of conjugacy classes of primitive elementsof F n . For z P F n , let v z w denote the union of the conjugacy classes of z and z ´ . A primitiveelement of F n is an element that forms part of a free basis. Let C “ tv z w | z P F n primitive u .The set C forms the set of vertices of a simplicial complex called the complex of partialbases which is analogous to the complex of curves for the mapping class group. Applying atheorem of the second author [31] to the action of IA n on the complex of partial bases, wewill obtain a decomposition IA n “ ˚ c P C p IA n q c {p some relations q ; (2)here p IA n q c denotes the stabilizer in IA n of c . The unlisted relations play only a small rolein our proof and can be ignored at this point. Step 2.
We use induction to construct a partial inverse.
Fix some c P C . The stabilizer p IA n q c is very similar to IA n ´ ; in fact, it is connected toIA n ´ by an exact sequence that is analogous to the Birman exact sequence for the mappingclass group. We construct this exact sequence in the companion paper [17], which buildson our previous paper [15]. By analyzing this exact sequence and using induction, we willconstruct a “partial inverse” φ c : p IA n q c Ñ Q n . We remark that this step is where most ofour relations arise—they actually are relations in the kernel of the Birman exact sequencewe construct in [15]. Step 3.
We use the L-presentation to lift the conjugation action of
Aut p F n q on IA n to Q n . Let r E be the induced endomorphisms of our L-presentation. We directly prove that theseendomorphisms actually give an action of Aut p F n q on Q n such that the projection map π : Q n Ñ IA n is equivariant. This is the key place where we use properties of L-presentations;in general, it is difficult to construct group actions on groups given by generators andrelations. Step 4.
We use our group action to construct the inverse.
The conjugation action of Aut p F n q on IA n transitively permutes the terms of (2). Using ourlifted action of Aut p F n q on Q n as a “guide”, we then “move” the partially defined inverse φ c around and construct φ on the rest of IA n , completing the proof.7 emark . In [30], the second author constructed an infinite presentation of the Torellisubgroup of the mapping class group. Though this used the same result [31] that we quotedabove, the details are quite different. One source of this difference is that instead of an L-presentation with a finite generating set, the paper [30] constructed an ordinary presentationwith an infinite generating set. In our paper [18], we construct a different presentation forIA n which is in the same spirit as the presentation in [30]. Computer calculations.
At several places in this paper, we will need to verify largenumbers of equations in group presentations. Rather than displaying these equations in thepaper or leaving them as exercises, we use the GAP System to store and check our equationsmechanically. The code to verify these equations is in the file h2ia.g , which is distributedwith this document and is also available on the authors’ websites. It is also attached tothis paper’s arXiv posting. We found the equations in this file by hand, and our proof doesnot rely on a computer search. We will say more about this in §
5, where said calculationsbegin.This is a good place to note that our results rely strongly on the authors’ earlier paper [17]and the computer calculutions from that paper. In that paper, we use a similar approachto automatically verify identities and prove the existence of certain homomorphisms tobetween groups given by presentations. This is used in more than one place in the presentpaper, but most crucially in Proposition 3.14. The computations from our earlier paper arein a file iabes.g , which is also available on the authors’ websites and on the arXiv.
Outline.
We begin in § § §
4. This proof depends on somecombinatorial group theory calculations that are stated in § § §
5. In §
6, we prove Theorem A. That section also shows how to deriveTheorem B from Theorem A. Finally, Theorems C and Theorem D are proven in § We now discuss the relations R p n q and endomorphisms E IA p n q of our L-presentation. Twocalculations (Propositions 2.1 and 2.3) are postponed until § Relations.
Our set R p n q of relations consists of the relations in Table 2. It is easy toverify that these are all indeed relations: Proposition 2.1.
The relations R p n q all hold when interpreted in IA n . The proof is computational and is postponed until §
5. We remark that unlike many of ourcomputer calculations, it is not particularly difficult to verify by hand.Since the relations are rather complicated we suggest to the reader that they not pay tooclose attention to them on their first pass through the paper. The overall structure of our8 asic relations for IA n Distinct letters are assumed to represent distinct indices unless stated otherwise. Let R IA p n q denote the finite set of all relations from the following ten classes.R0. M ´ x αa , r x βb ,x γc s “ M x αa , r x γc ,x βb s .R1. r C x a ,x b , C x c ,x d s “
1, possibly with b “ d .R2. r M x αa , r x βb ,x γc s , M x δd , r x ǫe ,x ζf s s “
1, possibly with t b, c u X t e, f u ‰ ∅ or with x αa “ x ´ δd ,as long as x αa ‰ x δd , a R t e, f u and d R t b, c u .R3. r C x a ,x b , M x γc , r x δd ,x ǫe s s “
1, possibly with b P t d, e u , if c R t a, b u and a R t c, d, e u .R4. r C x a ,x b C x c ,x b , C x c ,x a s “ βx a ,x b M x αa , r x βb ,x γc s C ´ βx a ,x b “ M x αa , r x γc ,x ´ βb s .R6. M x αa , r x βb ,x γc s M x ´ αa , r x βb ,x γc s “ r C ´ γx a ,x c , C ´ βx a ,x b s .R7. r C ´ βx a ,x b , M x βb , r x γc ,x δd s s “ r C ´ δx a ,x d , C ´ γx a ,x c s .R8. M x αa , r x βb ,x γc s M x δd , r x αa ,x ǫe s M x αa , r x γc ,x βb s “ C ´ ǫx d ,x e M x δd , r x γc ,x βb s C ǫx δd ,x e M x δd , r x αa ,x ǫe s M x δd , r x βb ,x γc s , possibly with b “ e or c “ e .R9. C βx a ,x b M x γc , r x αa ,x δd s C ´ β x a ,x b “ C ´ δx c ,x d M x γc , r x αa ,x βb s C δx c ,x d M x γc , r x αa ,x δd s M x γc , r x βb ,x αa s , possiblywith b “ d . Table 2:
Relations for the L-presentation of IA n . proof (and, in fact, the majority of its details) can be understood without much knowledgeof our relations. Remark . The relations in R p n q have reasonable intuitive interpretations. R1 throughR3 state that generators acting only in different places commute with each other. R4 is ageneralization of the fact that for n “
3, the conjugation move C x ,x conjugates the innerautomorphism C x ,x C x ,x back to itself (since it fixes the conjugating element x ). R5makes sense by looking at either side of x a : on the right of x αa , instances of x ˘ βb cancel,but on the left side of x αa , we get a conjugate of a basic commutator that is itself a basiccommutator. R6 states that conjugation by a commutator is the same as acting by acommutator of conjugation moves. R7, R8 and R9 allow us to rewrite a conjugate of agenerator acting on a given element as a product of generators acting only on that sameelement ( x a , x d or x c as stated here, respectively). In this sense, these relations are like theSteinberg relations from the presentation of GL n p Z q in algebraic K-theory. Generators for automorphism group of free group.
Before discussing our endomor-phisms E IA p n q , we first introduce a generating set for Aut p F n q that goes back to work ofNielsen. For α “ ˘ ď i, j ď n , let M x αi ,x j P Aut p F n q be the transvection x αi to x j x αi and fixes x ℓ for ℓ ‰ i . Just like before, we haveM x ´ i ,x j p x ´ i q “ x j x ´ i and M x ´ i ,x j p x i q “ x i x ´ j . Next, for distinct 1 ď i, j ď n let P i,j P Aut p F n q be the swap automorphism that exchanges x i and x j while fixing x ℓ for ℓ ‰ i, j . Finally, for 1 ď i ď n let I i P Aut p F n q be the inversionautomorphism that takes x i to x ´ i and fixes x ℓ for ℓ ‰ i . Define S Aut p n q “t M βx αi ,x j | ď i, j ď n distinct, α, β P t˘ uuY t P i,j | ď i, j ď n distinct u Y t I i | ď i ď n u . Observe that the set S Aut p n q Ă Aut p F n q is closed under inversion. Endomorphisms.
Below we will define a function θ : S Aut p n q Ñ End p F p S IA p n qqq withthe following key property. Let π : F p S IA p n qq Ñ IA n and ρ : F p S Aut p n qq Ñ Aut p F n q be theprojections. Then for s P S Aut p n q and w P F p S IA p n qq , we have π p θ p s qp w qq “ ρ p s q π p w q ρ p s q ´ P IA n . (3)Our set of endomorphisms will then be E IA p n q “ t θ p s q | s P S Aut p n qu . The relevance of the formula (3) is that we want the induced endomorphisms of our IA-presentation of IA n to generate the image of Aut p F n q in Aut p IA n q Ă End p IA n q arising fromthe conjugation action of Aut p F n q on IA n . Defining θ . Consider s P S Aut p n q . To define an endomorphism θ p s q : F p S IA p n qq Ñ F p S IA p n qq , it is enough to say what θ p s q does to each element of S IA p n q . There are twocases. • s “ P i,j or s “ I i . We then define θ p s q using the action of s on F n via formulas θ p s qp C x a ,x b q “ C s p x a q ,s p x b q and θ p s qp M x αa , r x βb ,x γc s q “ M s p x αa q , r s p x βb q ,s p x γc qs . Here one should interpret C x ´ e ,x f as C x e ,x f , C x e ,x ´ f as C ´ x e ,x f , and C x ´ e ,x ´ f as C ´ x e ,x f . • s “ M x αa ,x b . In this case, we define θ p s q via the formulas in Table 3. These list the caseswhere θ p s q does not fix a generator, except that to avoid redundancy, we do not alwayslist both a commutator transvection and its inverse. Specifically, if t “ M x γc , r x δd ,x ǫe s ,possibly with t c, d, e u X t a, b u ‰ ∅ , and a formula is listed for t “ M x γc , r x ǫe ,x δd s but notfor t , then we define θ p s qp t q “ θ p s qp t q ´ . If Table 3 lists no entry for t or t , or the table lists no entry for t and t is a conjugationmove, then we define θ p s qp t q “ t .These formulas were chosen to be as simple as possible, among formulas realizing equa-tion (3). Just like for the relations, we recommend not dwelling on these formulas duringone’s first read through this paper. 10 P S I θ p M βx αa ,x b qp s q C x c ,x a p C αx c ,x a C βx c ,x b q α C x a ,x c C x a ,x c M x αa , r x ´ βb ,x c s C x b ,x c C x b ,x c M x αa , r x ´ βb ,x ´ c s C x b ,x a p C βx a ,x b C αx b ,x a q α M x αa , r x γc ,x δd s C βx a ,x b M x αa , r x γc ,x δd s C ´ βx a ,x b M x γc , r x αa ,x δd s M x γc , r x βb ,x δd s C ´ βx c ,x b M x γc , r x αa ,x δd s C βx c ,x b M x γc , r x ´ αa ,x δd s M x γc , r x ´ αa ,x δd s C αx c ,x a M x γc , r x ´ βb ,x δd s C ´ αx c ,x a M x βb , r x γc ,x δd s C βx a ,x b M x ´ αa , r x γc ,x δd s M x βb , r x γc ,x δd s C ´ βx a ,x b M x ´ βb , r x γc ,x δd s M x αa , r x γc ,x δd s M x ´ βb , r x γc ,x δd s M x αa , r x βb ,x γc s C βx a ,x b M x αa , r x βb ,x γc s C ´ βx a ,x b M x αa , r x ´ βb ,x γc s C βx a ,x b M x αa , r x ´ βb ,x γc s C ´ βx a ,x b M x βb , r x αa ,x γc s C γx a ,x c M x αa , r x ´ βb ,x γc s C ´ γx b ,x c M x ´ βb , r x αa ,x ´ γc s M x βb , r x ´ αa ,x γc s C ´ βx c ,x b C ´ αx c ,x a M x ´ βb , r x ´ γc ,x αa s C γx b ,x c M x αa , r x γc ,x ´ βb s C ´ γx a ,x c C αx c ,x a C βx c ,x b M x ´ βb , r x αa ,x γc s C ´ γx a ,x c C αx c ,x a M x ´ βb , r x γc ,x ´ αa s C βx γc ,x b M x αa , r x γc ,x ´ βb s C γx b ,x c C ´ βx c ,x b C ´ αx c ,x a M x ´ βb , r x ´ αa ,x γc s C ´ γx b ,x c M x αa , r x ´ βb ,x γc s C ´ βx c ,x b M x ´ βb , r x ´ αa ,x γc s C ´ αx c ,x a C γx a ,x c C αx c ,x a C βx c ,x b M x γc , r x αa ,x βb s C βx a ,x b M x γc , r x αa ,x βb s C ´ βx a ,x b M x γc , r x αa ,x ´ βb s M x γc , r x βb ,x αa s Table 3:
Definition of θ p M βx αa ,x b q on the generators S IA p n q . All indices in each entry are assumedto be distinct. If no entry is listed for t P S IA p n q or for the generator representing t ´ (as in relationR0) then θ p M βx αa ,x b qp t q “ t . Proposition 2.3.
The definition of θ satisfies equation (3) . This proof uses a computer verification and is postponed until §
5. Propositions 2.1 and 2.3together imply that all of the extended relations from our L-presentation are trivial in IA n .This means that the obvious map on generators (sending each generator to the automor-phism it names) extends to a well defined homomorphism x S IA p n q | R p n q | E IA p n qy Ñ IA n . In this section, we assemble the tools we will need to prove Theorem E. In § § § n -stabilizers of simplices in the complex of partial bases. In § p F n q on the group given by our purported L-presentation for IA n .Finally, in § §
5: Propo-sition 3.13 from § § Consider a group G acting on a simplicial complex X . We say that G acts without rotations if for all simplices σ of X , the setwise and pointwise stabilizers of σ coincide. For a simplex σ , denote by G σ the stabilizer of σ . Letting X p q denote the vertex set of X , there is ahomomorphism from the free product of vertex stabilizers ψ : ˚ v P X p q G v ÝÑ G. As notation, if g P G stabilizes a vertex v of X , then denote by g v the associated element of G v ă ˚ v P X p q G v . The map ψ is rarely injective. Two families of elements in its kernel are as follows. • If e is an edge of X joining vertices v and v and if g P G e , then g v g ´ v P ker p ψ q . Wecall these the edge relators . • If v, w P X p q and g P G v and h P G w , then h w g v h ´ w p hgh ´ q ´ h p v q P ker p ψ q . We callthese the conjugation relators .The second author gave hypotheses under which these generate ker p ψ q . Theorem 3.1 ([31]) . Consider a group G acting without rotations on a -connected simpli-cial complex X . Assume that X { G is -connected. Then the kernel of the map ψ describedabove is normally generated by the edge and conjugation relators. We now introduce the simplicial complex to which we will apply Theorem 3.1. For z P F n ,let v z w denote the union of the conjugacy classes of z and z ´ . Definition 3.2. A partial basis for F n is a set t z , . . . , z k u Ă F n such that there exist z k ` , . . . , z n P F n with t z , . . . , z n u a free basis for F n . The complex of partial bases for F n , denoted B n , is the simplicial complex whose p k ´ q -simplices are sets tv z w , . . . , v z k wu ,where t z , . . . , z k u is a partial basis for F n .The group Aut p F n q acts on B n , and we wish to apply Theorem 3.1 to the restriction of thisaction to IA n . It is clear that IA n acts on B n without rotations, so we must check that B n is 1-connected and that B n { IA n is 2-connected.We start by verifying that B n is 1-connected. Proposition 3.3.
The simplicial complex B n is -connected for n ě . roof. For z P F n , let v z w be the conjugacy class of z . Define B n to be the simplicialcomplex whose p k ´ q -simplices are sets tv z w , . . . , v z k w u , where t z , . . . , z k u is a partialbasis for F n . In [16], the authors proved that B n is 1-connected for n ě
3. There is a naturalsimplicial map ρ : B n Ñ B n . Letting ψ : p B n q p q Ñ p B n q p q be an arbitrary map satisfying ρ ˝ ψ “ id, it is clear that ψ extends to a simplicial map ψ : B n Ñ B n satisfying ρ ˝ ψ “ id.This implies that ρ induces a surjection on all homotopy groups, so B n is 1-connected for n ě B n { IA n is p n ´ q -connected. In particular, it is 2-connectedfor n ě
4, and thus satisfies the conditions of Theorem 3.1 for n ě
4. However, we willneed a complex that satisfies the conditions of Theorem 3.1 for n “ Definition 3.4.
The augmented complex of partial bases for F n , denoted p B n , is the simpli-cial complex whose p k ´ q -simplices are as follows. • Sets of the form tv z w , . . . , v z k wu , where t z , . . . , z k u is a partial basis for F n . Thesewill be called the standard simplices . • Sets of the form tv z z w , v z w , v z w , . . . , v z k ´ wu , where t z , . . . , z k ´ u is a partial basisfor F n . These will be called the additive simplices . Remark . Since z z and z z are conjugate, the two additive simplices tv z z w , v z w , v z w , . . . , v z k ´ wu and tv z z w , v z w , v z w , . . . , v z k ´ wu of p B n are the same.The group Aut p F n q (and hence IA n ) still acts on p B n . Since p B n is obtained from B n by addingsimplices of dimension at least 2, it inherits the 1-connectivity of B n for n ě Proposition 3.6.
The complex p B n is -connected for n ě . To help us understand the connectivity of p B n { IA n , we introduce the following complex. For ~v P Z n , let p ~v q ˘ denote the set t ~v, ´ ~v u . Definition 3.7. A partial basis for Z n is a set t ~v , . . . , ~v k u Ă Z n such that there exist ~v k ` , . . . , ~v n P Z n with t ~v , . . . , ~v n u a basis for Z n . The augmented complex of lax partialbases for Z n , denoted p B n p Z q , is the simplicial complex whose p k ´ q -simplices are as follows. • Sets of the form tp ~v q ˘ , . . . , p ~v k q ˘ u , where t ~v , . . . , ~v k u is a partial basis for Z n . Thesewill be called the standard simplices . • Sets of the form tp ~v ` ~v q ˘ , p ~v q ˘ , p ~v q ˘ , . . . , p ~v k ´ q ˘ u , where t ~v , . . . , ~v k ´ u is a partialbasis for Z n . These will be called the additive simplices .We then have the following lemma. Lemma 3.8.
We have p B n { IA n – p B n p Z q for n ě . For the proof of Lemma 3.8, we will need the following result of the authors. For z P F n ,let r z s P Z n be the associated element of the abelianization of F n .13 emma 3.9 ([16, Lemma 5.3]) . Let t ~v , . . . , ~v n u be a basis for Z n and let t z , . . . , z k u be apartial basis for F n such that r z i s “ ~v i for ď i ď k . Then there exists z k ` , . . . , z n P F n with r z i s “ ~v i for k ` ď i ď n such that t z , . . . , z n u is a basis for F n .Proof of Lemma 3.8. The map p p B n q p q Ñ p p B n p Z qq p q that takes v z w to r z s extends to asimplicial map ρ : p B n Ñ p B n p Z q . Since IA n acts without rotations on p B n , the quotient p B n { IA n has a natural CW-complex structure whose k -cells are the IA n -orbits of the k -cells of p B n (warning: though it will turn out that in this case it is, this CW-complexstructure need not be a simplicial complex structure; consider, for example, the action of Z by translations on the standard triangulation of R whose vertices are Z ). Since ρ isIA n -invariant, it factors through a map ρ : p B n { IA n Ñ p B n p Z q . We will prove that ρ is anisomorphism of CW-complexes.This requires checking two things. The first is that every simplex of p B n p Z q is in the imageof ρ , which is an immediate consequence of Lemma 3.9. The second is that if σ and σ are simplices of p B n such that ρ p σ q “ ρ p σ q , then there exists some f P IA n such that f p σ q “ σ . It is clear that σ and σ are either both standard simplices or both additivesimplices. Assume first that they are both standard simplices. We can then write σ “ tv z w , . . . , v z k wu and σ “ tv z w , . . . , v z k wu as in the definition of standard simplices with r z i s “ r z i s for 1 ď i ď k . Set ~v i “ r z i s “ r z i s for 1 ď i ď k . The set t ~v , . . . , ~v k u is a partial basis for Z n , so we can extend it to a basis t ~v , . . . , ~v n u . Applying Lemma 3.9 twice, we can find z k ` , . . . , z n P F n and z k ` , . . . , z n P F n such that r z i s “ r z i s “ ~v i for k ` ď i ď n and such that both t z , . . . , z n u and t z , . . . , z n u are free bases for F n . There then exists f P Aut p F n q such that f p z i q “ z i for 1 ď i ď n . Byconstruction, we have f P IA n and f p σ q “ σ .It remains to deal with the case where σ and σ are both simplices of additive type. Write σ “ tv z z w , v z w , v z w , . . . , v z k ´ wu and σ “ tv z z w , v z w , . . . , v z k ´ wu as in the definition of additive simplices. The unordered sets tpr z s`r z sq ˘ , pr z sq ˘ , pr z sq ˘ u and tpr z s`r z sq ˘ , pr z sq ˘ , pr z sq ˘ u are minimal nonempty subsets of ρ p σ q “ ρ p σ q such thatthe defining elements of Z n are not linearly independent. It follows that as unordered setswe have ρ ptv z z w , v z w , v z wuq “ ρ ptv z z w , v z w , v z wuq and ρ ptv z w , . . . , v z k ´ wuq “ ρ ptv z w , . . . , v z k ´ wuq . Reordering the z i and possibly replacing some of the z i by z ´ i (which does not change v z i w ),we can assume that r z i s “ r z i s for 3 ď i ď k ´ tv z z w , v z w , v z wu , tv z z w , v z w , v z wu , tv z w , v z z w , v z ´ wu , tv z ´ w , v z w , v z ´ z ´ wu , tv z w , v z z w , v z ´ wu , tv z ´ w , v z w , v z ´ z ´ wu .
14y reordering σ and possibly changing some of our expressions for the elements in it again,we can assume that pr z s ` r z sq ˘ “ pr z s ` r z sq ˘ and pr z sq ˘ “ pr z sq ˘ and pr z sq ˘ “ pr z sq ˘ and that r z i s “ r z i s for 3 ď i ď k ´ pr z s , r z sq “ pr z s , r z sq or pr z s , r z sq “ p´r z s , ´r z sq ;the key point here is that changing the sign of one of tr z s , r z su but not the other changes pr z s ` r z sq ˘ . If the second possibility occurs, then replace z and z with z ´ and z ´ ,respectively; this does not change σ . The upshot is that we now have arranged for r z i s “ r z i s for all 1 ď i ď k ´
1. By the same argument we used to deal with standard simplices, thereexists some f P IA n such that f p z i q “ z i for 1 ď i ď k ´
1. Since f p z z q “ z z , we seethat f p σ q “ σ , as desired.The second author together with Church proved in [13] that p B n p Z q is p n ´ q -connected for n ě
1. We therefore deduce the following.
Proposition 3.10.
The complex p B n { IA n is p n ´ q -connected for n ě . This section is devoted to the following proposition, which gives generators for the stabilizersin IA n of simplices of B n . Recall that S MA p n q is Magnus’s generating set for IA n discussedin the introduction. Proposition 3.11.
Fix ď k ď n and define Γ “ p IA n q v x n ´ k ` w , v x n ´ k ` w ,..., v x n w . Then Γ isgenerated by S MA p n q X Γ “ t C x a ,x b | ď a, b ď n distinct uY t M x a , r x b ,x c s | ď a ď n ´ k , ď b, c ď n distinct u . Proof.
The map F n Ñ F n ´ k that quotients out by the normal closure of t x n ´ k ` , . . . , x n u induces a split surjection ρ : Γ Ñ IA n ´ k . Define K n ´ k,k “ ker p ρ q , so we have Γ “ K n ´ k,k ¸ IA n ´ k . As we said in the introduction, Magnus [28] proved that IA n ´ k is generated by t C x a ,x b | ď a, b ď n ´ k distinct u Y t M x a , r x b ,x c s | ď a, b, c ď n ´ k distinct u . (4)The authors proved in [17, Theorem A] that K n ´ k,k is generated by t C x a ,x b | n ´ k ` ď a ď n , 1 ď b ď n distinct uY t C x a ,x b | ď a ď n , n ´ k ` ď b ď n distinct uY t M x a , r x b ,x c s | ď a ď n ´ k , n ´ k ´ ď b ď n , 1 ď c ď n distinct u . (5)The union of (4) and (5) is the claimed generating set for Γ. Remark . For z P F n , define v z w to be the conjugacy class of z . The reference [17] actu-ally deals with p IA n q v x n ´ k ` w , v x n ´ k ` w ,..., v x n w instead of p IA n q v x n ´ k ` w , v x n ´ k ` w ,..., v x n w ; however,since x i and x ´ i have different images in F ab n these two stabilizer subgroups are actuallyequal. There are also notational differences: the group denoted K n ´ k,k here is denoted K IA n ´ k,k in that paper. 15 .4 The action of Aut p F n q Let Q n be the group with the L-presentation x S IA p n q | R p n q | E IA p n qy discussed in § π : Q n Ñ IA n . The group Aut p F n q actson IA n by conjugation. The goal of this section is to state Proposition 3.13 below, whichasserts that this action can be lifted to Q n .To state some important properties of this lifted action, we must introduce some notation.First, let S Aut p n q Ă Aut p F n q be the generating set discussed in §
2. Recall that E IA p n q Ă End p F p S IA p n qqq is the image of a map θ : S Aut p n q Ñ End p F p S IA p n qqq . There is thus amap e : S Aut p n q Ñ End p Q n q whose image is the set of induced endomorphisms of our L-presentation. It will turn out that the image of e consists of automorphisms and theseautomorphisms generate the action of Aut p F n q on Q n .Second, recall that t x , . . . , x n u is a fixed free basis for F n . Let p S IA p n qq v x n w “t C x a ,x b | ď a, b ď n distinct uY t M x αa , r x βb ,x γc s | ď a, b, c ď n distinct, α, β, γ P t˘ u , a ‰ n u . This is exactly the subset of S IA p n q Ă IA n consisting of automorphisms that fix v x n w ;Proposition 3.11 (with k “
1) implies that it generates the stabilizer subgroup p IA n q v x n w .Define p Q n q v x n w be the subgroup of Q n generated by p S IA p n qq v x n w . We will then require thestabilizer subgroup p Aut p F n qq v x w to preserve the subgroup p Q n q v x n w .Our proposition is as follows. Proposition 3.13.
For all n ě , there is an action of Aut p F n q on Q n that satisfies thefollowing three properties.1. The action comes from the induced endomorphisms in the sense that for s P S IA p n q Ă Aut p F n q and q P Q n , we have s ¨ q “ e p s q ¨ q .2. The restriction of the action to IA n induces the conjugation action of Q n on itself inthe sense that for q, r P Q n , we have π p r q ¨ q “ rqr ´ .3. For η P p
Aut p F n qq v x n w and q P p Q n q v x n w , we have η ¨ q P p Q n q v x n w . The proof of Proposition 3.13 is a computation with generators and relations (mostly doneby computer), so we have postponed it until § There is a natural split surjection ρ : p IA n q v x n w Ñ IA n ´ arising from the map F n Ñ F n ´ which quotients out the normal closure of x n . Let K n ´ , “ ker p ρ q , so we have a decompo-sition p IA n q v x n w “ K n ´ , ¸ IA n ´ . Building on the Birman exact sequence for Aut p F n q weconstructed in [15], we constructed an L-presentation for K n ´ , in [17, Theorem D] ( K n ´ , is denoted K IA n ´ , in that paper). This L-presentation plays a crucial role in the inductivestep of our proof, because it allows us to obtain the following proposition:16 roposition 3.14. There is a homomorphism K n ´ , Ñ x S IA p n q | R p n q | E IA p n qy fittinginto the following commuting triangle: K n ´ , / / v(cid:22) ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ x S IA p n q | R p n q | E IA p n qy (cid:15) (cid:15) (cid:15) (cid:15) IA n Usually finding a homomorphism between groups given by presentations is simple: onechecks that the relations map to products of conjugates of relations. This is the spirit ofthe proof of Proposition 3.14, but the substitution rules and extended relations complicatethe picture. Our proof of Proposition 3.14 is computer-assisted and is postponed until § In this section, we prove Theorem E, which says that IA n has the finite L-presentation x S IA p n q | R p n q | E IA p n qy discussed in §
2. Our proof is inspired by the proof of the maintheorem of [9]. We will make use of Propositions 2.1, 2.3, 3.13, and 3.14, which were allstated in previous sections and which will be proved (with the aid of a computer) in § Proof of Theorem E.
As notation, let Q n be the group given by x S IA p n q | R p n q | E IA p n qy .Elements of S IA p n q play dual roles as elements of Q n and as elements of IA n , and during ourproof it will be important to distinguish them. Therefore, throughout this proof elementsC x a ,x b and M x αa , r x βb ,x γc s will always lie in IA n ; the associated elements of Q n will be denoted C x a ,x b and M x αa , r x βb ,x γc s .There is a natural projection map π : Q n Ñ IA n . We will prove that π is an isomorphismby induction on n . The base cases are n “ n “
2. For n “
1, both IA n and Q n are the trivial group, so there is nothing to prove. For n “
2, it is a classical theorem ofNielsen ([29]; see also [27, Proposition 4.5]) that IA is the group of inner automorphismsof F , so IA is a free group on the generators C x ,x and C x ,x . Our generating set for Q is t C x ,x , C x ,x u , and for n “
2, the set of basic relations R p q is empty. Even thoughour set of substitution rules E IA p q is nonempty, it follows that our full set of relations for Q is empty. So our presentation for Q is x C x ,x , C x ,x | Hy , and the result is also truein this case.Assume now that n ě Q n Ñ IA n is an isomorphism forall 1 ď n ă n . Since π is a surjection, to prove that π is an isomorphism it is enough toconstruct a homomorphism φ : IA n Ñ Q n such that φ ˝ π “ id. Propositions 3.6 and 3.10show that the action of IA n on p B n satisfies the conditions of Theorem 3.1, soIA n – ˜ ˚ v z wPp p B n q p q p IA n q v z w ¸ { R, where R is the normal closure of the edge and conjugation relators. The construction of φ will have two steps. First, we will use the action of Aut p F n q on Q n provided by Proposition17.13 to construct a map r φ : ˚ v z wPp p B n q p q p IA n q v z w ÝÑ Q n . Second, we will show that r φ takes the edge and conjugation relators to 1, and thus inducesa map φ : IA n Ñ Q n . We will close by verifying that φ ˝ π “ id. Construction of r φ . To construct r φ , we must construct a map r φ v z w : p IA n q v z w ÝÑ Q n for each vertex v z w of p B n . Recalling that t x , . . . , x n u is our fixed free basis for F n , webegin with the vertex v x n w . In the following claim, we will use the notation p S IA p n qq v x n w and p Q n q v x n w introduced in § Claim 1.
The restriction of π to p Q n q v x n w is an isomorphism onto p IA n q v x n w .Proof of claim. Proposition 3.11 implies that natural map π | p Q n q v xn w : p Q n q v x n w Ñ p IA n q v x n w is surjective, since the generators from that proposition (with k “
1) are in the image.Our inductive hypothesis says that the map π | Q n ´ : Q n ´ Ñ IA n ´ is an isomorphism.Recall from § p IA n q v x n w “ K n ´ , ¸ IA n ´ , where the projection p IA n q v x n w Ñ IA n ´ is the one induced by the map F n Ñ F n ´ that quotients out by the normal closure of x n ,and K n ´ , is the kernel of this projection. The composition p Q n q v x n w ÝÑ p IA n q v x n w ÝÑ IA n ´ π | ´ Q n ´ ÝÑ – Q n ´ is a well defined homomorphism. It is a composition of surjective maps, and is thereforesurjective. We define K n ´ , to be the kernel of this composition of maps.The restriction of π to K n ´ , has its image in K n ´ , since the map p Q n q v x n w Ñ Q n ´ factors through p IA n q v x n w Ñ IA n ´ . Proposition 3.11 says that K n ´ , is generated by theset S K p n q : “t C x n ,x a , C x a ,x n | ď a ă b uY t M x αa , r x βb ,x γn s , M x αa , r x γn ,x βb s | ď a, b ă n distinct, α, β, γ P t˘ uu . Since these generators are contained in K n ´ , , the map K n ´ , Ñ K n ´ , is surjective.Further, Proposition 3.14 gives us a left inverse to π | K n ´ , . We conclude that π | K n ´ , is anisomorphism K n ´ , – K n ´ , . We note that existence of this isomorphism is a deceptivelydifficult part of the proof, and it is the main consequence that we draw from [17].Summing up, we have a commutative diagram of short exact sequences as follows.1 ÝÝÝÝÑ K n ´ , ÝÝÝÝÑ p Q n q v x n w ÝÝÝÝÑ Q n ´ ÝÝÝÝÑ §§đ – §§đ §§đ – ÝÝÝÝÑ K n ´ , ÝÝÝÝÑ p IA n q v x n w ÝÝÝÝÑ IA n ´ ÝÝÝÝÑ p Q n q v x n w Ñ p IA n q v x n w is an isomor-phism, as desired. 18laim 1 implies that we can define a map r φ v x n w : p IA n q v x n w Ñ Q n via the formula r φ v x n w “p π | p Q n q v xn w q ´ .Now consider a general vertex v z w of p B n . Here we will use the action of Aut p F n q on Q n provided by Proposition 3.13. The group Aut p F n q acts transitively on the set of primitiveelements of F n , so there exists some ν P Aut p F n q such that ν p x n q “ z . We then define amap r φ v z w : p IA n q v z w Ñ Q n via the formula r φ v z w p η q “ ν ¨ r φ v x n w p ν ´ ην q p η P p IA n q v z w q . This appears to depend on the choice of ν , but the following claim says that this choicedoes not matter. Claim 2.
The map r φ v z w p η q does not depend on the choice of ν .Proof of claim. Assume that ν , ν P Aut p F n q both satisfy ν i p x n q “ z , and consider some η P p IA n q v z w . Our goal is to prove that ν ¨ r φ v x n w p ν ´ ην q “ ν ¨ r φ v x n w p ν ´ ην q . (6)Define µ “ ν ´ ν and ω “ ν ´ ην , so µ P p
Aut p F n qq v x n w and ω P p IA n q v x n w . We will firstprove that r φ v x n w p µωµ ´ q “ µ ¨ r φ v x n w p ω q . (7)To see this, observe first that by construction both r φ v x n w p µωµ ´ q and r φ v x n w p ω q lie in p Q n q v x n w .The third part of Proposition 3.13 implies that µ ¨ r φ v x n w p ω q also lies in p Q n q v x n w . Claim 1says that π | p Q n q v xn w is injective, so to prove (7), it is thus enough to prove that r φ v x n w p µωµ ´ q and µ ¨ r φ v x n w p ω q have the same image under π . This follows from the calculation π p r φ v x n w p µωµ ´ qq “ µωµ ´ “ µπ p r φ v x n w p ω qq µ ´ “ π p µ ¨ r φ v x n w p ω qq , where the first two equalities follow from the fact that π ˝ r φ v x n w “ id and the third followsfrom the first conclusion of Proposition 3.13.We now verify (6) as follows: ν ¨ r φ v x n w p ν ´ ην q “ ν ¨ r φ v x n w p µωµ ´ q “ ν µ ¨ r φ v x n w p ω q “ ν ¨ r φ v x n w p ν ´ ην q . This completes the construction of r φ . Some naturality properties.
Before we study the edge and conjugation relators, wefirst need to verify the following two naturality properties of r φ . Starting now we will usethe following notation which was introduced in §
2: for a vertex v z w of p B n and η P IA n satisfying η pv z wq “ v z w , we will denote η considered as an element of p IA n q v z w ă ˚ v z wPp p B n q p q p IA n q v z w by η v z w . 19 laim 3. The following two identities hold. • Let ď a, b ď n be distinct and ď i ď n be arbitrary. Then r φ pp C x a ,x b q v x i w q “ C x a ,x b . • Let ď a, b, c ď n be distinct, let α, β, γ P t˘ u be arbitrary, and let ď i ď n besuch that i ‰ a . Then r φ pp M x αa , r x βb ,x γc s q v x i w q “ M x αa , r x βb ,x γc s .Proof of claim. The proofs of two two identities are similar; we will deal with the first andleave the second to the reader. It is clear from the construction that r φ pp C x a ,x b q v x n w q “ C x a ,x b .For 1 ď i ă n , we have P i,n p x n q “ x i , and thus by definition we have r φ pp C x a ,x b q v x i w q “ P i,n ¨ r φ v x n w p P ´ i,n C x a ,x b P i,n q “ P i,n ¨ r φ v x n w p C P ´ i,n p x a q ,P ´ i,n p x b q q“ P i,n ¨ C P ´ i,n p x a q ,P ´ i,n p x b q “ C x a ,x b ;here the last equality follows from the first part of Proposition 3.13 and the definition ofthe endomorphisms in § Claim 4.
Let v z w be a vertex of p B n . Then for η P p IA n q v z w we have π p r φ v z w p η qq “ η .Proof of claim. Pick ν P Aut p F n q such that ν p x n q “ z . Then π p r φ v z w p η qq “ π p ν ¨ r φ v x n w p ν ´ ην qq “ νπ p r φ v x n w p ν ´ ην qq ν ´ “ νν ´ ηνν ´ “ η ;here the second equality uses the first part of Proposition 3.13. The edge and conjugation relators.
We now check that r φ takes the edge and conju-gation relators to 1. Claim 5 (Edge relators) . If e is an edge of p B n with endpoints v z w and v z w and η P p IA n q e ,then r φ p η v z w η ´ v z w q “ .Proof of claim. We first consider the special case where z “ x n and z “ x n ´ . Proposi-tion 3.11, with k “
2, states that p IA n q v x n ´ w , v x n w is generated by t C x a ,x b | ď a, b ď n distinct u Y t M x a , r x b ,x c s | ď a, b, c ď n distinct, a ‰ n ´ , n u . (8)Claim 3 implies that for all elements ω in (8), we have r φ p ω v x n ´ w q “ r φ p ω v x n w q . It followsthat for all η P p IA n q v x n ´ w , v x n w we have r φ p η v x n ´ w q “ r φ p η v x n w q , as desired.We now turn to general edges e with endpoints v z w and v z w and η P p IA n q e . There existssome ν P Aut p F n q such that ν p x n q “ z and ν p x n ´ q “ z , and hence νP n ´ ,n p x n q “ z .Setting η “ ν ´ ην P p IA n q v x n ´ w , v x n w , we have r φ p η v z w q “ ν ¨ r φ v x n w p ν ´ ην q “ ν ¨ r φ v x n w p η q and r φ p η v z w q “ νP n ´ ,n r φ v x n w p P ´ n ´ ,n ν ´ ηνP n ´ ,n q “ ν ¨ r φ v x n ´ w p η q . By the previous paragraph, we have r φ v x n w p η q “ r φ v x n ´ w p η q , so we conclude that r φ p η v z w q “ r φ p η v z w q , as desired. 20 laim 6 (Conjugation relators) . If v z w and v z w are vertices of p B n and η P p IA n q v z w and ω P p IA n q v z w , then r φ p ω v z w η v z w ω ´ v z w p ωηω ´ q v ω p z qw q “ . Proof of claim.
Choose ν P Aut p F n q such that ν p x n q “ z . We then have r φ p ω v z w η v z w ω ´ v z w q “ r φ v z w p ω q r φ v z w p η q r φ v z w p ω q ´ “ π p r φ v z w p ω qq ¨ r φ v z w p η q “ ω ¨ r φ v z w p η q“ ων ¨ r φ v x n w p ν ´ ην q “ ων ¨ r φ v x n w pp ων q ´ ωηω ´ p ων qq “ r φ pp ωηω ´ q v ω p z qw q , as desired. The second equality follows from the third part of Proposition 3.13, the thirdequality follows from Claim 4, and the remainder of the equalities are straightforwardapplications of the definitions.Claims 5 and 6 imply that r φ descends to a homomorphism φ : IA n Ñ Q n . We have an inverse.
To complete the proof, it remains to prove the following.
Claim 7.
We have φ ˝ π “ id.Proof of claim. Claim 3 implies that this holds for the generators of Q n .This completes the proof of Theorem E. This section contains the postponed proofs of Propositions 2.1, 2.3, 3.13, and 3.14. Theseproofs are done with the aid of a computer. We will discuss our computational frameworkin § § S Aut p n q ˚ denote thefree monoid on the set S Aut p n q . In §
2, we defined a function θ : S Aut p n q Ñ End p F p S IA p n qqq .This naturally extends to a function θ : S Aut p n q ˚ Ñ End p F p S IA p n qqq . As we discussed in the introduction, we use the GAP System to mechanically verify thelarge number of equations we have to check. These verifications are in the file h2ia.g ,distributed with this document and also available at the authors’ websites.We use GAP’s built-in functionality to model F n as a free group on the eight generators xa , xb , xc , xd , xe , xf , xg , and y . Since our computations never involve more than 8 variables,computations in this group suffice to show that our computations hold in general.Elements of the sets S Aut p n q and S IA p n q are parametrized over basis elements from F n andtheir inverses, so we model these sets using lists. For example, we model the generator M x a ,x b as the list ["M",xa,xb] , C y,x a as ["C",y,xa] , and M x ´ a , r y,x c s as ["Mc",xa^-1,y,xc] . We21odel P a,b as ["P",xa,xb] and I a as ["I",xa] . The examples should make clear: the firstentry in the list is a string key "M" , "C" , "Mc" , "P" , or "I" , indicating whether the listrepresents a transvection, conjugation move, commutator transvection, swap or inversion.The parameters given as subscripts in the generator are then the remaining elements of thelist, in the same order.GAP’s built-in free group functionality expects the basis elements to be variables, not lists,so we do not use it to model S Aut p n q ˚ and F p S IA p n qq . We model inverses of generatorsas follows: the inverse of ["M",xa,xb] is ["M",xa,xb^-1] and the inverse of ["C",xa,xb] is ["C",xa,xb^-1] , but the inverse of ["Mc",xa,xb,xc] is ["Mc",xa,xc,xb] . Swaps andinversions are their own inverses. Technically, this means that we are not really modeling S Aut p n q ˚ and F p S IA p n qq ; instead we model structures where the order relations for swapsand inversions and the relation R0 for inverting commutator transvections are built in.This is not a problem because our verifications always show that certain formulas are trivialmodulo our relations, and we can always apply the R0 and order relations as needed.We model words in S Aut p n q ˚ and F p S IA p n qq as lists of generators and inverse generators.The empty word [] represents the trivial element. We wrote several functions in h2ia.g thatperform common tasks on words. The function pw takes any number of words (reduced ornot) as arguments and returns the freely reduced product of those words in the given order,as a single word. The function iw inverts its input word and the function cyw cyclicallypermutes its input word.The function iarel outputs the relations R p n q . We introduce some extra relations forconvenience. The function exiarel outputs these extra relations and the code generatingthe list exiarelchecklist derives the extra relations from the basic relations. The function theta takes in a word w in S Aut p n q and a word v in S IA p n q , and returns θ p w qp v q . Inaddition to the functions described here, we often define simple macros for carrying out theverifications.The function applyrels is particularly useful, because it inserts multiple relations into aword. It takes in two inputs: a starting word and a list of words with placement indicators.The function recursively inserts the first word from the list in the starting word at the givenposition, reduces the word, and then calls itself with the new word as the starting word andwith the same list of insertions, with the first dropped.For example, the following command appears in the justification of Proposition 3.13: applyrels(pw( theta([["M",xa,xb],["M",xa,xb^-1]], [["Mc",xb,xa,xe]]),[["Mc",xb,xe,xa]]),[ [6,iarel(5,[xa,xb,xe])],[6,iw(exiarel(1,[xb^-1,xe,xa]))],[2,iw(iarel(4,[xe,xa,xb]))],[2,iarel(6,[xb^-1,xe^-1,xa])],[2,iw(iarel(5,[xb,xe,xa]))] ) This tells the GAP system to compute the effect of θ p M x a ,x b M ´ x a ,x b q on M x b , r x a ,x e s . Thenit multiplies this by M x b , r x e ,x a s , the inverse of M x b , r x a ,x e s . It then freely reduces this word.The system inserts a version of R5 after the sixth letter in this word, and reduces the resultto a new word. Then it inserts the inverse of one of the extra relations after the sixthletter in the new word and reduces it. It continues with inserting relations and reducingthe resulting expressions, inserting instances of R4, R5 and R6. Since the entire expressionevaluates to [] , we have expressed θ p M x a ,x b M ´ x a ,x b qp M x b , r x a ,x e s q ¨ M x b , r x e ,x a s as a product of relations in Q n . In any example like this, an interested reader can reproduceour reduction process by removing all the list entries from the second input of the applyrels call, and then adding them back in one at a time, evaluating after each one. IA n First we prove Proposition 2.1, which states that our relations R p n q hold in IA n . Proof of Proposition 2.1.
The code generating the list verifyiarel generates examples ofall the relations in R p n q , with all allowable configurations of coincidences between thesubscripts on the generators. It converts each of these relations into automorphisms of F n and evaluates them on a basis for F n , returning true if all basis elements are unchanged.We evaluate on a fixed finite-rank free group, but since the basic relations involve at mostsix generators, those evaluations suffice to show the result in general. Since verifyiarel evaluates to a list of true , this means that all these relations are true.Next we prove Proposition 2.3, which states that θ acts by conjugation when evaluated ongenerators (see equation (3)). Proof of Proposition 2.3.
The code generating the list thetavsconjaut goes through allpossible configurations for a pair of generators s from S Aut p n q and t from S IA p n q , evaluates θ p s qp t q as a product of generators, and then evaluates both θ p s qp t q and sts ´ on a basisfor F n . It returns true when both have the same effect on all basis elements. Since thetavsconjaut evaluates to a list of true , the proposition holds. Proof of Proposition 3.13.
The action of Aut p F n q on Q n is given by our substitution ruleendomorphism map θ : S Aut p n q Ñ End p F p S IA p n qqq . s P S Aut p n q , the element θ p s q defines an endomorphismof Q n . This is because the subgroup of F p S IA p n qq normally generated by the relations of Q n is invariant under θ p s q by the definition of Q n .Next, we verify that θ p s q is an automorphism of Q n . If s is a swap or an inversion, thenit is clear from the definition of θ that this is the case. In the code generating the list thetainverselist , we compute θ p s qp θ p s ´ qp t qq t ´ for s “ M x a ,x b and for all possible con-figurations of t relative to s . In each case, we reduce it to the trivial word using relationsfor Q n . It is not hard to deduce that θ p s qp θ p s ´ qp t qq “ t in Q n for the remaining choices of s “ M ´ x a ,x b , M x ´ a ,x b and M ´ x ´ a ,x b , using the fact that it is true for s “ M x a ,x b .So this shows that θ defines an action F p S Aut p n qq Ñ Aut p Q n q . Now we need to verify that this action descends to an action of Aut p F n q . To show this, itis enough to show that for every relation r in a presentation for Aut p F n q , we have θ p r qp t q “ t in Q n , (9)for t taken from a generating set for Q n . To check this, we use the same version of Nielsen’spresentation for Aut p F n q that we used in [17, Theorem 5.5]. The generators are the sameset S Aut p n q we use here, and the relations fall into four classes N1–N5. N1 are sufficientrelations for the subgroup generated by swaps and inversions and N2 are relations indicatinghow to conjugate transvections by swaps and inversions. It is an exercise to see that (9)holds for relations of class N1 and N2. Relations N3, N4 and N5 are more complicatedrelations. For each of these, we compute θ p r qp t q t ´ on generators t (for t with enoughconfigurations of subscripts to include a generating set) and reduce the resulting expressionsto 1 using relations from Q n . These computations are given in the code generating the lists thetaN3list , thetaN4list , and thetaN5list . Since these evaluate to lists of the trivialword, this verifies equation (9). We have shown that the action of an element of Aut p F n q on Q n does not depend on the word in F p S Aut p n qq we use to represent it.Since we have shown that θ defines an action, now we can check the three properties assertedin Proposition 3.13. We have already verified the first point (we took the definition of theaction to agree with it). To verify the second point, we need to check that for w, s P S IA p n q ,there is r w P F p S Aut p n qq representing w with θ p r w qp s q “ wsw ´ in Q n . (10)In fact, it is enough to verify this for w, s in a smaller generating set, and the generatingset that s is taken from may depend on w . In the code generating thetaconjrellist , foreach choice of w from S MA p n q , we lift w to r w P F p S Aut p n qq , and for several configurationsof subscripts in the generator s , we reduce θ p r w qp s q ws ´ w ´ to the identity using relationsfrom Q n . We use enough configurations of subscripts in s to cover all cases for s in agenerating set (a conjugate of S MA p n q ).To check the third point, we use the generating set p S Aut p n qq v x n w for p Aut p F n qq v x n w men-tioned in the proof of Proposition 3.11 above, namely t M x αa ,x b | ď a ď n ´ ď b ď n , α “ ˘ a ‰ b u Y t P a,b | ď a, b ď n , a ‰ b , a ‰ n uY t I a | ď a ď n u Y t C x n ,x a | ď a ď n ´ u .
24e need to check that for each of these generators, there is w P F p S Aut p n qq representing itwith θ p w qp s q in p Q n q v x n w (really, that θ p w qp s q is equal in Q n to an element of p Q n q v x n w ). Thisis clear from the definition of θ for w a swap or inversion. It can be verified for w “ M x αa ,x b by inspecting Table 3. For w representing C x n ,x a , the fact that θ p w qp s q P p Q n q v x n w followsfrom the second point in this proposition, since C x n ,x a P IA n . Here we prove Proposition 3.14. We recall the statement: K n ´ , is the kernel of the naturalmap p IA n q v x n w Ñ IA n ´ , and the proposition asserts that the inclusion K n ´ , ã Ñ IA n factorsas the composition of a map K n ´ , Ñ Q n with the projection Q n Ñ IA n .The proof uses the finite L-presentation for K n ´ , from [17]. We note that [17, Theorem D]asserts the existence of such a presentation, and [17, Theorem 6.2] gives the precise state-ment that we use in the computations. Since this L-presentation is in fact a presentationfor K n ´ , , we use the same notation for K n ´ , as a subset of IA n and K n ´ , as the groupgiven by this presentation.We do not reproduce the L-presentation here, but instead we describe some of its features.Its finite generating set is S K p n q “t M x αa , r x ǫn ,x βb s | ď a, b ď n ´ a ‰ b , α, β, ǫ P t , ´ uuY t C x n ,x a | ď a ď n ´ u Y t C x a ,x n | ď a ď n ´ u . The substitution endomorphisms of the L-presentation for K n ´ , are indexed by a finitegenerating set p S Aut p n qq v x n w for p Aut p F n qq v x n w . The endomorphisms themselves are theimage of a map φ : p S Aut p n qq v x n w Ñ End p F p S K p n qqq . Proof of Proposition 3.14.
Since S K p n q is a subset of S IA p n q , we map K n ´ , to Q n bysending each generator to the generator of the same name. To verify that this map ongenerators extends to a well defined map of groups, we need to check that each definingrelation from K n ´ , maps to the trivial element of Q n . Since K n ´ , is given by a L-presentation, we proceed as follows:1. We check that each of the basic relations from K n ´ , maps to the trivial element of Q n .2. We check that for s P p S Aut p n qq v x n w and t P S K p n q , we have φ p s qp t q “ θ p s qp t q in Q n , where we use p S Aut p n qq v x n w Ă S Aut p n q to plug s into θ , and we interpret both expres-sions in Q n using F p S K p n qq Ă F p S IA p n qq .The first point is verified in the code generating the list kfromialist . The function krel produces the basic relations from K n ´ , , and we reduce each relation to the identity byapplying relations from Q n . The second point is verified in the code generating the list25 hetavsphlist . For each choice of pairs of generators, we reduce the difference of θ and φ using relations from Q n .With these two points verified, one can easily check by induction that every extendedrelation (starting with a basic relation, and applying any sequence of rewriting rules) mapsto the identity element in Q n . H p IA n q In this section, we prove Theorem A, which asserts that there exists a finite subset ofH p IA n q whose GL n p Z q -orbit spans H p IA n q . In fact, we gave an explicit list of generators inTable 1; each generator is of the form h r for a commutator relation r . This list is reproducedin Table 4, which also introduces the notation h i p . . . q P H p IA n q for the associated elementsof homology (this notation will be used during the calculations in §
7, though we will notuse it in this section). The following theorem asserts that this list is complete; it is a moreprecise form of Theorem A and will be the main result of this section.
Theorem 6.1.
Fix n ě . Let S H p n q be the set of commutator relators in Table 4. Thenthe GL n p Z q -orbit of the set t h r | r P S H p n qu spans H p IA n q . Before proving Theorem 6.1, we will use it to derive Theorem B.
Proof of Theorem B.
Recall that this theorem asserts that for n ě
6, the GL n ` p Z q -orbitof the image of the natural map H p IA n q Ñ H p IA n ` q spans H p IA n ` q . Let S n ` Ă GL n ` p Z q be the subgroup consisting of permutation matrices. By inspecting Table 4, itis clear that the S n ` -orbit of the image of t h r | r P S H p n qu Ă H p IA n q in H p IA n ` q is t h r | r P S H p n ` qu . This uses the fact that n ě
6, since the commutator relations in S H p n q use generators involving at most six basis elements.We now turn to the proof of Theorem 6.1. We start by introducing some notation. Let F “ F p S IA p n qq and let R Ă F denote the full set of relations of IA n , so IA n “ F { R . Define Č H p IA n q “ R {r F, R s , and for r P R denote by } r } the associated element of Č H p IA n q . Thereis a natural map Č H p IA n q Ñ F ab , and the starting point for our proof is the followinglemma. In it, recall from the beginning of § S Aut p n q ˚ is the free monoid on the set S Aut p n q . Lemma 6.2.
The group Č H p IA n q is an abelian group which is generated by t} θ p w qp r q} | w P S Aut p n q ˚ and r is one of the relations R0–R9 from Table 2 u . Also, we have H p IA n q “ ker p Č H p IA n q Ñ F ab q .Proof. The group Č H p IA n q is abelian since r R, R s Ă r
F, R s . For v P F and r P R , we have r v, r s P r F, R s , so } vrv ´ } “ } r } . The indicated generating set for Č H p IA n q thus followsfrom Theorem E. As for the final statement of the lemma, we follow one of the standard26roofs of Hopf’s formula [10]. The 5-term exact sequence in group homology associated tothe short exact sequence 0 ÝÑ R ÝÑ F ÝÑ IA n ÝÑ p F q ÝÑ H p IA n q ÝÑ R {r F, R s ÝÑ H p F q ÝÑ H p IA n q ÝÑ . Since F is free, we have H p F q “
0, and the claim follows.Our goal will be to take an element of Č H p IA n q that happens to lie in H p IA n q and rewriteit as a sum of elements of the form t} θ p w qp r q} | w P S Aut p n q ˚ and r is one of the relations H1–H9 from Table 4 u . (11)The relations H1–H4 are the same as R1–R4, and the relations H5–H7 are the same asR6–R9. The troublesome relations are R0, R5, and R6, none of which lie in H p IA n q . For r P R , we have } r } P H p IA n q if and only if the exponent-sum of each generator in S IA p n q appearing in it is 0. For our problematic relations R0, R5, and R6, the exponent-sum ofall the conjugations moves is already 0, so we will only need to study the exponent-sums ofthe commutator transvections.We begin with the following lemma, which will allow us to mostly ignore our rewriting rules θ p¨q . Lemma 6.3.
Consider w P S Aut p n q ˚ , and let r be a relation of the form R0, R5, or R6.Then } θ p w qp r q} “ h ` h , where h and h are as follows. • h P H p IA n q is a sum of elements from (11) . • h is a sum of elements of the form t} r } | r is one of the relations R0, R5, and R6 from Table 2 u . Proof.
We can use induction to reduce to the case where w “ s P S Aut p n q . The proof nowis a combinatorial group-theoretic calculation: we will show how to rewrite θ p s qp r q as aproduct of relations of the desired form.We start by dealing with the case where r is of the form R0. Observe that θ p s qp M x αa , r x βb ,x γc s q agrees with θ p s qp M x αa , r x γc ,x βb s q ´ up to R0, except in two cases. These are θ p M βx αa ,x b qp M x ´ βb , r x γc ,x δd s q “ M x αa , r x γc ,x δd s M x ´ βb , r x γc ,x δd s and θ p M βx αa ,x b qp M x βb , r x γc ,x δd s q “ C βx a ,x b M x ´ αa , r x γc ,x δd s M x βb , r x γc ,x δd s C ´ βx a ,x b . In the first case, θ p M βx αa ,x b qp M x ´ βb , r x δd ,x γc s q ´ “ M x ´ βb , r x γc ,x δd s M x αa , r x γc ,x δd s . In the second case, θ p M βx αa ,x b qp M x βb , r x δd ,x γc s q ´ “ C βx a ,x b M x βb , r x γc ,x δd s M x ´ αa , r x γc ,x δd s C ´ βx a ,x b . asic commutator relators for IA n H1. h p x αa , x βb , x γc , x δd q : possibly with b “ d , r C βx a ,x b , C δx c ,x d s “ . H2. h p x αa , x βb , x γc , x δd , x ǫe , x ζf q : possibly with t b, c u X t e, f u ‰ ∅ or with x αa “ x ´ δd , aslong as x αa ‰ x δd , a R t e, f u and d R t b, c u , r M x αa , r x βb ,x γc s , M x δd , r x ǫe ,x ζf s s “ . H3. h p x γc , x δd , x ǫe , x αa , x βb q : possibly with b P t d, e u , if c R t a, b u and a R t c, d, e u , r C βx a ,x b , M x γc , r x δd ,x ǫe s s “ . H4. h p x αa , x βb , x γc q : r C βx c ,x b C βx a ,x b , C αx c ,x a s “ . H5. h p x αa , x βb , x γc , x δd q : r C ´ γx a ,x c , C ´ δx a ,x d sr C ´ βx a ,x b , M x βb , r x γc ,x δd s s “ . H6. h p x αa , x βb , x γc , x δd , x ǫe q : possibly with b “ e or c “ e , r M x αa , r x βb ,x γc s , M x δd , r x αa ,x ǫe s sr M x δd , r x αa ,x ǫe s , M x δd , r x γc ,x βb s sr M x δd , r x γc ,x βb s , C ´ ǫx d ,x e s “ . H7. h p x αa , x βb , x γc , x δd q : possibly with b “ d , r M x γc , r x αa ,x δd s , C βx a ,x b sr C ´ δx c ,x d , M x γc , r x αa ,x βb s sr M x γc , r x αa ,x βb s , M x γc , r x αa ,x δd s s “ . H8. h p x αa , x βb , x γc , x δd q : r M x αa , r x βb ,x γc s , C δx a ,x d C δx b ,x d C δx c ,x d s “ . H9. h p x αa , x βb , x γc q : r C γx a ,x c C γx b ,x c , C βx a ,x b C βx c ,x b sr M x αa , r x βb ,x γc s , C αx b ,x a C αx c ,x a s “ . Table 4:
The set S H p n q of commutator relators such that the GL n p Z q -orbit of t h r | r P S H p n qu spans H p IA n q . Distinct letters are assumed to represent distinct indices unless stated otherwise.We give notation h i p . . . q for the elements in H p IA n q , which we use later.
28n both cases, the two expressions differ by an application of R2. This means that θ p s q ofan R0 relation can always be written using R0 and R2 relations.Next we explain the computations that prove the lemma for R5 and R6 relations. These arein the list rewritethetaR5R6 . In these computations we reduce θ p s qp r q to the trivial wordwhere s P S ˘ A and r is an R5 or R6 relation. These reductions may use any of the basicrelations for IA n , including R5 and R6 themselves, but notably may not use the images ofR5 and R6 relations under θ . We may use the images of R1–R4, R7–R9, H8 and H9 under θ .Despite these restrictions, we may use the extra relations from exiarel in these compu-tations. Our relation exiarel(3,[xa,xb,xc,xd]) is H8, and exiarel(4,[xa,xb,xc,xd]) is equivalent to H8 modulo the basic relations. Relation exiarel(7,[xa,xb,xc]) is H9,and relation exiarel(6,[xa,xb,xc]) and relation exiarel(8,[xa,xb,xc,xd]) are equiv-alent to H9 modulo the basic relations. All the other exiarel relations can be derivedwithout using images of R5 or R6 under θ . These facts can be verified by inspecting exiarelchecklist .If s is a swap or an inversion move, then acting by θ p s q is always the same as acting onthe parameters of the relation by s in the obvious way. Therefore rewritethetaR5R6 onlycontains cases where s is a transvection.We use several redundancies between different forms of the relations R5 and R6 to reducethe number of computations. Inverting the parameter x βb in R5 (as it appears in Table 2)is the same as cyclically permuting the relation. Inverting the parameter x αa in R6 is thesame as applying a relation from R2 to the original R6 relation. Swapping the roles of x βb and x γc in R6 is the same as inverting and cyclically permuting the original R6 relation andapplying a relation from R2.We use the identity: C ´ βx a ,x b θ p M βx αa ,x b qp t q C βx a ,x b “ θ p M ´ βx ´ αa ,x b qp t q which holds in IA n for any t P IA n . This is a consequence of Proposition 3.13. In particular,this means that we only need to consider one of θ p M x αa ,x βb qp r q and θ p M x ´ αa ,x ´ βb qp r q ; if one istrivial then so is the other.Since the computations in rewritethetaR5R6 rewrite all configurations of θ p s qp r q for r anR5 or R6 relation, up to these reductions, this proves the lemma.The next lemma allows us to deal with certain combinations of R5 and R6 relations. Theordered triple of generators of F n involved in a commutator transvection M x αi , r x βj ,x γk s is p x i , x j , x j q . There are eight commutator transvections involving a given triple of generators. Lemma 6.4.
Fix distinct ď a, b, c ď n , and let w P R X r
F, F s be a product of R5 and R6relations whose commutator transvections involve only p x a , x b , x c q , in order. Then } w } canbe written as a sum of elements of the form } v } with v an H2 relation.Proof. Let F be the subgroup of F generated by the eight commutator transvections in-volving p x a , x b , x c q and the two conjugation moves t C x a ,x b , C x a ,x c u , and let R Ă F bethe normal closure in F of the R5 and R6 relations that can be written as products of29ame Generator v M x a , r x b ,x c s v M x ´ a , r x b ,x c s v M x a , r x ´ b ,x c s v M x ´ a , r x ´ b ,x c s v M x a , r x b ,x ´ c s v M x ´ a , r x b ,x ´ c s v M x a , r x ´ b ,x ´ c s v M x ´ a , r x ´ b ,x ´ c s Name Relation r C x a ,x b M x a , r x b ,x c s C ´ x a ,x b M x a , r x ´ b ,x c s r C x a ,x b M x ´ a , r x b ,x c s C ´ x a ,x b M x ´ a , r x ´ b ,x c s r C ´ x a ,x c M x a , r x b ,x c s C x a ,x c M x a , r x b ,x ´ c s r C ´ x a ,x c M x ´ a , r x b ,x c s C x a ,x c M x ´ a , r x b ,x ´ c s r C ´ x a ,x c M x a , r x ´ b ,x c s C x a ,x c M x a , r x ´ b ,x ´ c s r C ´ x a ,x c M x ´ a , r x ´ b ,x c s C x a ,x c M x ´ a , r x ´ b ,x ´ c s r C x a ,x b M x a , r x b ,x ´ c s C ´ x a ,x b M x a , r x ´ b ,x ´ c s r C x a ,x b M x ´ a , r x b ,x ´ c s C ´ x a ,x b M x ´ a , r x ´ b ,x ´ c s r M x a , r x b ,x c s M x ´ a , r x b ,x c s r C ´ x a ,x b , C ´ x a ,x c s r M x a , r x ´ b ,x c s M x ´ a , r x ´ b ,x c s r C x a ,x b , C ´ x a ,x c s r M x a , r x b ,x ´ c s M x ´ a , r x b ,x ´ c s r C ´ x a ,x b , C x a ,x c s r M x a , r x ´ b ,x ´ c s M x ´ a , r x ´ b ,x ´ c s r C x a ,x b , C x a ,x c s r M x ´ a , r x b ,x c s M x a , r x b ,x c s r C ´ x a ,x b , C ´ x a ,x c s r M x ´ a , r x ´ b ,x c s M x a , r x ´ b ,x c s r C x a ,x b , C ´ x a ,x c s r M x ´ a , r x b ,x ´ c s M x a , r x b ,x ´ c s r C ´ x a ,x b , C x a ,x c s r M x ´ a , r x ´ b ,x ´ c s M x a , r x ´ b ,x ´ c s r C x a ,x b , C x a ,x c s Table 5:
Labels for the eight commutator transvections using x a , x b and x c in order, and for thesixteen R5 and R6 relations using these commutator transvections. elements of F . We thus have w P R X r F , F s . Our first goal is to better understand p R X r F , F sq{r F , R s and R {r F , R s . Consider the exact sequence of abelian groups0 ÝÑ R X r F , F sr F , R s ÝÑ R r F , R s ÝÑ R R X r F , F s ÝÑ . We find generators for the first group in the sequence by considering a related exact sequenceof free abelian groups.Let v , . . . , v be the eight commutator transvections in S IA p n q that only involve p x a , x b , x c q ,enumerated as in Table 5. Similarly, let r , . . . , r be the eight R5 relations in F and let r , . . . , r denote the eight R6 relations lying in F , enumerated as in Table 5. Let A bethe free abelian group freely generated by r , . . . , r , and let B be the free abelian groupfreely generated by v , . . . , v . We consider the map A Ñ B that counts the exponent-sumof each commutator transvection generator. Let C denote the kernel of this map and let B denote the image.Since R is normally generated by relations r , . . . , r , we know that R {r F , R s is generatedby the images of these relations. Thus there is a surjection A Ñ R {r F , R s that sends eachbasis element to the image of the relation with the same name. The group B is a subgroup of F {r F , F s . The natural map R {r F , R s Ñ F {r F , F s counts exponent-sums of generators.Since the generators r , . . . , r all have zero exponent-sum for conjugation move generators,we do not lose any information by counting only commutator transvection generators in B .30his means we have a commuting square A / / (cid:15) (cid:15) B (cid:15) (cid:15) R r F ,R s / / F r F ,F s . The subgroup B thus maps surjectively onto p R r F , F sq{r F , F s , which is isomorphic to R {p R X r F , F sq . Therefore we have a commuting diagram with exact rows0 / / C / / (cid:15) (cid:15) A / / (cid:15) (cid:15) B / / (cid:15) (cid:15) / / R Xr F ,F sr F ,R s / / R r F ,R s / / R R Xr F ,F s / / B Ñ F {r F , F s is injective, so B Ñ R {p R X r F , F sq is also injective. Byconstruction, A Ñ R {r F , R s is surjective. It follows from a simple diagram chase that C Ñ p R X r F , F sq{r F , R s is surjective.The map A Ñ B is given by this 8 ˆ
16 matrix: ¨˚˚˚˚˚˚˚˚˚˚˝ ˛‹‹‹‹‹‹‹‹‹‹‚
A straightforward linear algebra computation shows that C , the kernel of this map, isgenerated by the following nine vectors: r ´ r ´ r ` r , r ´ r ´ r ` r , ´ r ´ r ` r ` r ´ r ´ r ` r ` r , r ` r ´ r ´ r ´ r ` r ,r ´ r , r ´ r , r ´ r , and r ´ r . Since C surjects on p R Xr F , F sq{r F , R s , we know that p R Xr F , F sq{r F , R s is generatedby the images of these nine elements.We will now describe calculations that show that each of the generators above is equivalentmodulo r F , R s to an H2 relation. In each case, we find representatives in R Xr F , F s of theimage of the given element of C . Since we are working modulo r F , R s we may conjugateany r i in computing the representative. In reducing to an H2 relation, we may also applyany relation at all, as long as we apply its inverse somewhere else.The last four generators are easily equivalent to H2 relations. We skip the second kernelgenerator because its image is equal to the first after inverting x a , and we skip the fourthbecause because its image is equal to the third after swapping x b and x c . The three com-putations in the list kernellist finish the lemma by showing that the first, third and fifthgenerators are equivalent to H2 relations. 31 roof of Theorem 6.1. We must show that every element of Č H p IA n q that happens to lie inH p IA n q can be written as a sum of elements of t} θ p w qp r q} | w P S Aut p n q ˚ and r is one of the relations H1–H9 from Table 4 u . Combining Lemmas 6.2 and 6.3 with the fact that R0 and R5 and R6 are the only relationsin our L-presentation for IA n that do not appear as one of the commutator relations inTable 4, we see that it enough to deal with sums of elements of the set t} r } | r is one of the relations R0, R5, and R6 u . So consider } w } P H p IA n q that can be written } w } “ m ÿ i “ } r i } with each r i either an R0 or R5 or R6 relation.For any choice of distinct 1 ď a, b, c ď n , we consider the commutator transvection gener-ators involving p x a , x b , x c q or p x a , x c , x b q , and the R5, R6 and R0 relations involving onlythese commutator transvections. We write } w } “ n ` p n ´ q ÿ i “ } w i } , where each } w i } is a sum of R5, R6 and R0 relations involving only a single choice of p x a , t x b , x c uq . To prove the theorem, it is enough to show that we can write each of the } w i } as a sum of our generators for H p IA n q . So we assume } w } “ } w i } for some i ; this amountsto fixing a choice of p x a , t x b , x c uq and assuming } w } is a sum of R0, R5 and R6 generatorsusing only commutator transvections involving this triple.We call a commutator transvection M x αi , r x βj ,x γk s positive if j ă k , and negative otherwise.Each R5 or R6 relation contains two positive commutator transvections (and no negativeones), or two negative ones (and no positive ones). Suppose } r i } is an R5 or R6 relationwith negative generators, appearing in the sum defining } w } . By inserting an R0 relationand its inverse into r i , we replace both of the negative generators with positive ones. Let r i denote the word we get by doing this to r i . Since we have added and subtracted thesame element in Č H p IA n q , we have } r i } “ } r i } . Modifying an R5 or R6 relation in this waygives us the inverse of an R5 or R6 relation involving the same p x a , t x b , x c uq , up to cyclicpermutation of the relation. So we interpret this move as rewriting the sum defining } w } :we replace the relation } r i } with the new relation } r i } , which is an R5 or R6 relation withoutnegative commutator transvections. We proceed to eliminate all the negative commutatortransvections in R5 and R6 relations in } w } this way.Having done this, the only negative commutator transvections the sum defining } w } appearin R0 relations. Since } w } P H p IA n q , the negative generators appear with exponent-sumzero; so the R0 relations appear in inverse pairs. This means that we can simply rewritethe sum without any R0 relations. So } w } is a sum of R5 and R6 relations whose onlycommutator transvections are positive ones involving p x a , t x b , x c uq . Then } w } satisfies thehypotheses of Lemma 6.4 and therefore is a sum of H2 generators.32 Coinvariants and congruence subgroups
This section contains the proofs of Theorems C and D, which can be found in § § GL n p Z q on H p IA n q This section is devoted to understanding the action of GL n p Z q on our generators for H p IA n q .The results in this section consist of long lists of equations that are verified by a computer,so on their first pass a reader might want to skip to the next two sections to see how theyare used. For i “ , . . . ,
9, we use the notation h i p x α a , . . . , x α ki a ki q P H p IA n q for the imagein H p IA n q of the i th relation from from S H p n q , with the given parameters, as specified inTable 4. Since the action of GL n p Z q is induced from the action of Aut p F n q , we record theaction of various Aut p F n q generators on these generators.The computations justifying Lemmas 7.2–7.6 are in the file h2ia.g . We use the Hopfisomorphism H p IA n q – p R X r
F, F sq{r
F, R s , where F “ F p S IA p n qq and R ă F is the groupof relations of IA n . We justify these equations by performing computations in R Xr F, F s Ă F .In each computation, we start with a word representing one side of the equation and reduceto the trivial word using words representing the other side. Since r F, R s is trivial, we mayuse any relations in inverse pairs, we may apply relations from in any order, and we maycyclically permute relations.We note some identities, which we leave as an exercise. Lemma 7.1.
The following identities hold in H p IA n q . The letters in subscripts are as-sumed distinct unless otherwise noted. • h p x αa , x βb , x γc , x δb q “ ´ h p x αa , x βb , x γc , x ´ δb q , even if b “ d . • h p x αa , x βb , x γc , x δb , x ǫe q “ ´ h p x αa , x βb , x γc , x δb , x ´ ǫe q , even if b “ e or c “ e . • h p x αa , x βb , x γc , x δb , x ǫe q “ ´ h p x αa , x γc , x βb , x δb , x ǫe q , even if b “ e or c “ e . We also need the following, which is not obvious.
Lemma 7.2.
The following identities hold in H p IA n q . The letters in subscripts are as-sumed distinct unless otherwise noted. • h p x αa , x ǫe , x γc , x δd , x ǫe q “ h p x αa , x γc , x ´ ǫe , x δd , x ǫe q´ h p x αa , x ǫe , x δd , x ǫe q ´ h p x αa , x ´ ǫe , x δd , x ǫe q . (12) • h p x αa , x βb , x γc , x δd , x ǫe q “ ´ h p x αa , x γc , x βb , x δd , x ǫe q , even if b “ e or c “ e . (13) Proof.
Computations justifying these equations appear in the list lemma7pt2 .We proceed by expressing the action of many elementary matrices from GL n p Z q on ourgenerators. Lemma 7.3.
The following identities hold in H p IA n q . The letters in subscripts are as-sumed distinct unless otherwise noted. M ǫx βb ,x e ¨ h p x αa , x βb , x γc , x δd q “ h p x αa , x βb , x γc , x δd q ` h p x αa , x ǫe , x γc , x δd q . (14) • M ǫx βb ,x e ¨ h p x αa , x ǫe , x γc , x δd q “ h p x αa , x ǫe , x γc , x δd q . (15) • M δx βb ,x d ¨ h p x αa , x βb , x γc , x δd q “ h p x αa , x βb , x γc , x δd q ` h p x αa , x δd , x γc , x δd q . (16) • M ´ ǫx αa ,x e ¨ h p x αa , x βb , x γc , x δd q “ h p x αa , x βb , x γc , x δd q ´ h p x αa , x ǫe , x βb , x γc , x δd q , (17) even if x ǫe “ x δd . • M ´ ǫx δd ,x e ¨ h p x αa , x βb , x γc , x δd , x ζf q “ h p x αa , x βb , x γc , x δd , x ζf q´ h p x αa , x βb , x γc , x δd , x ǫe , x ζf q , (18) even if t b, c u X t e, f u ‰ ∅ . • M δx ´ αa ,x d ¨ h p x αa , x βb , x γc , x ´ δd , x ζf , x ǫe q “ h p x αa , x βb , x γc , x ´ δd , x ζf , x ǫe q´ h p x αa , x βb , x γc , x ´ αa , x ǫe , x ζf q , (19) even if t b, c u X t e, f u ‰ ∅ .Proof. These computations appear in lemma7pt3 . The equations where coincidences areallowed are justified in several different computations.
Lemma 7.4.
The following identities hold in H p IA n q . The letters in subscripts are as-sumed distinct. • M βx δd ,x b ¨ h p x αa , x δd , x γc q “ h p x αa , x δd , x γc q ` h p x αa , x βb , x γc q . (20) • M βx δd ,x b ¨ h p x αa , x βb , x γc q “ h p x αa , x βb , x γc q . (21) • M ´ γx βb ,x c ¨ h p x αa , x βb , x γc , x δd q “ h p x αa , x βb , x γc , x δd q ´ h p x βb , x δd , x γc , x αa , x γc q` h p x αa , x βb , x γc , x δd q ` h p x γc , x δd , x αa q . (22) • M ǫx δd ,x e ¨ h p x αa , x βb , x γc , x δd q “ h p x αa , x βb , x γc , x δd q ` h p x αa , x βb , x γc , x ǫe q` (H1 generators). (23) • M ǫx δd ,x e ¨ h p x αa , x βb , x γc , x ǫe q “ h p x αa , x βb , x γc , x ǫe q . (24) Proof.
These computations appear in lemma7pt4 . Lemma 7.5.
The following identities hold in H p IA n q . The letters in subscripts are as-sumed distinct. • M ǫx ζf ,x e ¨ h p x αa , x βb , x γc , x δd , x ζf q “ h p x αa , x βb , x γc , x δd , x ζf q` h p x αa , x βb , x γc , x δd , x ǫe q . (25) • M ǫx ζf ,x e ¨ h p x αa , x βb , x γc , x δd , x ǫe q “ h p x αa , x βb , x γc , x δd , x ǫe q . (26) • M ´ ǫx γc ,x e ¨ h p x αa , x βb , x ǫe , x δd q “ h p x αa , x βb , x ǫe , x δd q ` h p x αa , x βb , x γc , x δd q` h p x ǫe , x βb , x αa , x γc , x δd q` (H1, H2, and H3 generators). (27) • M ´ ǫx γc ,x e ¨ h p x αa , x βb , x γc , x δd q “ h p x αa , x βb , x γc , x δd q` h p x ´ γc , x ´ ǫe , x ´ δd , x γc , x βb , x αa q . (28)34 M βx δd ,x b ¨ h p x αa , x δd , x γc , x δd q “ h p x αa , x δd , x γc , x δd q ` h p x αa , x βb , x γc , x βb q` h p x αa , x βb , x γc , x δd q ` h p x αa , x δd , x γc , x βb q` (H1 generators). (29) • M βx δd ,x b ¨ h p x αa , x βb , x γc , x βb q “ h p x αa , x βb , x γc , x βb q . (30) • M βx δd ,x b ¨ h p x αa , x ´ δd , x γc , x δd q “ h p x αa , x ´ δd , x γc , x δd q ` h p x αa , x ´ βb , x γc , x βb q` h p x αa , x ´ δd , x γc , x βb q ´ h p x αa , x ´ βb , x γc , x ´ δd q . (31) • M βx δd ,x b ¨ h p x αa , x ´ βb , x γc , x βb q “ h p x αa , x ´ βb , x γc , x βb q . (32) • M ǫx ζf ,x e ¨ h p x αa , x ζf , x γc , x δd , x ζf q “ h p x αa , x ζf , x γc , x δd , x ζf q ´ h p x αa , x ǫe , x δd , x ζf q´ h p x αa , x ǫe , x δd , x ǫe q ` h p x αa , x ζf , x γc , x δd , x ǫe q´ h p x αa , x ´ ǫe , x δd , x ζf q ´ h p x αa , x ´ ǫe , x δd , x ǫe q` h p x αa , x ǫe , x γc , x δd , x ζf q ` h p x αa , x ǫe , x γc , x δd , x ǫe q . (33) • M ǫx ζf ,x e ¨ h p x αa , x ǫe , x γc , x δd , x ǫe q “ h p x αa , x ǫe , x γc , x δd , x ǫe q . (34) Proof.
These computations appear in lemma7pt5 . Lemma 7.6.
The following identities hold in H p IA n q . The letters in subscripts are as-sumed distinct. • M δx βb ,x d ¨ h p x αa , x βb , x γc q “ h p x αa , x βb , x γc q ` h p x αa , x δd , x γc q` (H1–H5 generators). (35) • M δx βb ,x d ¨ h p x αa , x δd , x γc q “ h p x αa , x δd , x γc q ` (H1–H6 generators). (36) Proof.
These computations are contained in the list lemma7pt6 , but they take some expla-nation. We use the relations exiarel(5,[...]) frequently in these computations; theserelations are always combinations of H1 and H4 relations. We use exiarel(7,[xa,xb,xc]) to represent h p x αa , x βb , x γc q , but we also use other relations in this computation. The re-lation exiarel(8,[xa,xb,xc,xd]) is an expanded version of this relation that behavesbetter under this action. Its derivation in exiarelchecklist shows that it differs from ´ h p x αa , x βb , x γc q only by H1, H4 and H5 relations. We also use exiarel(6,[xa,xb,xc,xd]) ;this differs from h p x αa , x βb , x γc q by H1 and H4 relations, and an R6 relation. This is apparentin the derivation of exiarel(7,[xa,xb,xc]) in exiarelchecklist .The computation justifying equation (35) shows directly that the image of the relation exiarel(8,[xa,xb,xc,xd]) under [["M",xb,xd]] can be reduced to the trivial word byapplying: • exiarel(6,[xa,xd,xc]) , • iw(exiarel(7,[xa,xb,xc])) , • H1–H5 relations (some in exiarel(5,[...]) relations), • R5 and R6 relations, and • elements from r F, R s (including inverse pairs of instances of exiarel(1,[...]) and exiarel(3,[...]) ). 35ince we start and end with elements of R X r
F, F s in this computation, the use of R5 andR6 (in one case inside exiarel(6,[xa,xb,xc,xd]) ) is inconsequential; by Lemma 6.4 thiscan only change the outcome by H2 relations. So this proves equation (35).The computation justifying equation (36) is similar, but uses an instance of the relation exiarel(2,[...]) . The derivation in exiarelchecklist shows that this relation is acombination of H2, H5, and H6 relations, and elements of r F, R s . H p IA n q In this section, we prove Theorem C, which asserts that for the GL n p Z q -coinvariants ofH p IA n q vanish for n ě Proof of Theorem C.
We use the generators H1–H9 from Table 4. To show that the coin-variants H p IA n q GL n p Z q are trivial, we show that the coinvariance class of each of thesegenerators is trivial. The covariants are defined to be the largest quotient of H p IA n q witha trivial induced action of GL p n, Z q . Since the action of GL p n, Z q on H ˚ p IA n q is induced bythe action of Aut p F n q on IA n , this means that H p IA n q GL n p Z q is the quotient of H p IA n q bythe subgroup generated by classes of the form f ¨ r ´ r , where f P Aut p F n q and r P H p IA n q .Elements of the form f ¨ r ´ r are called coboundaries .In fact, we have already shown in Lemmas 7.1–7.6 that the subgroup of H p IA n q generatedby coboundaries contains our generators from Theorem 6.1. We show this for each generatorin the equations above as follows. Each equation shows how to express the given generatoras a sum of coboundaries and generators previously expressed in terms of coboundaries: • H1 in equations (14) and (16), also using an observation from Lemma 7.1, • H2 in equations (18) and (19), • H3 in equation (17), also using an observation from Lemma 7.1, • H4 in equation (20), • H5 in equation (22), • generic H6 in equation (25), • H7 in equations (27), (29), and (31), • the special cases of H6 in equation (33), also using equations (12) and (13), • H8 in equation (23) and • H9 in equation (35).
Remark . The equations above assume that distinct subscripts label distinct elements.This means that equation (25) requires six basis elements. We do not know if the generic H6generator (a five-parameter generator) can be expressed as a sum of coboundaries withoutusing a sixth basis element. Therefore we require n ě n . In this section, we prove Theorem D, which asserts that H p Aut p F n , ℓ q ; Q q “ n ě ℓ ě
2. The key to this is the following lemma. Let GL n p Z , ℓ q be the level- ℓ congruence36ubgroup of GL n p Z q , that is, the kernel of the natural map GL n p Z q Ñ GL n p Z { ℓ q .Like in Theorem C, we require n ě n . Lemma 7.8.
For n ě and ℓ ě we have p H p IA n ; Q qq GL n p Z ,ℓ q “ .Proof. Again we use our generators from Theorem 6.1. The universal coefficient theoremimplies that H p IA n ; Q q is generated by the images of our generators from H p IA n q .We have two approaches for showing that a generator has trivial image. The first isthe following: if f P Aut p F n q and r, s P H p IA n q with f ¨ r ´ r “ s and f ¨ s “ s in p H p IA n ; Q qq GL n p Z ,ℓ q , then f ℓ ¨ r ´ r “ ℓs in p H p IA n ; Q qq GL n p Z ,ℓ q . If further, f ℓ lies in Aut p F n , ℓ q , then this shows that s is trivial in p H p IA n ; Q qq GL n p Z ,ℓ q .The second approach is simpler: if f P Aut p F n q and r, s P H p IA n q with f ¨ r ´ r “ s and r “ p H p IA n ; Q qq GL n p Z ,ℓ q , then of course, s “ p H p IA n ; Q qq GL n p Z ,ℓ q .We show the generators have trivial images as follows: • generic H1 using equations (14) and (15) by the first approach, • special cases of H1 using equation (16) by the second approach, and using Lemma 7.1, • H3 by equation (17) using the second approach, and using Lemma 7.1, • H2 by equations (18) and (19), using the second approach, • H4 by equations (20) and (21), using the first approach, • H5 by equation (22), by the second approach, • generic H6 by equations (25) and (26), using the first approach, • generic H7 by equations (27) and (28), by the first approach, • special cases of H7 by equations (29) and (30), and by equations (31) and (32), bothby the first approach, • one special case of H6 by equations (33) and (34), by the first approach, • other special cases of H6 using the first case and equations (12) and (13), • H8 by equations (23) and (24) by the first approach and • H9 by equations (35) and (36) by the first approach.
Proof of Theorem D.
We examine the Hochschild-Serre spectral sequence associated to theshort exact sequence 1 ÝÑ IA n ÝÑ Aut p F n , ℓ q ÝÑ GL n p Z , ℓ q ÝÑ . (37)First, the Borel stability theorem [7] implies that H p GL n p Z , ℓ q ; Q q “
0. Next, recall fromthe introduction that H p IA n ; Q q – Hom p Q n , Ź Q n q ; the group GL n p Z q acts on this in theobvious way. Since Hom p R n , Ź R n q is an irreducible representation of the algebraic groupSL n p R q , it follows from work of Borel [6, Proposition 3.2] that H p IA n ; Q q is an irreduciblerepresentation of GL n p Z , ℓ q (we remark that the above reference is one of the steps in theoriginal proof of the Borel density theorem; the result can also be derived directly from the37orel density theorem). It thus follows from the extension of the Borel stability theorem tonontrivial coefficient systems [8] thatH p GL n p Z , ℓ q ; H p IA n ; Q qq “ . Lemma 7.8 says thatH p GL n p Z , ℓ q ; H p IA n ; Q qq – p H p IA n ; Q qq GL n p Z ,ℓ q “ . The p ` q “ p Aut p F n , ℓ q ; Q q “
0, as desired.
References [1] S. Andreadakis, On the automorphisms of free groups and free nilpotent groups, Proc. London Math.Soc. (3) 15 (1965), 239–268.[2] O. Baker, The Jacobian map on Outer Space, thesis, Cornell University, 2011.[3] L. Bartholdi, Endomorphic presentations of branch groups, J. Algebra 268 (2003), no. 2, 419–443.[4] M. Bestvina, K.-U. Bux and D. Margalit, Dimension of the Torelli group for Out p F n q , Invent. Math.170 (2007), no. 1, 1–32. arXiv:math/0603177 .[5] S. K. Boldsen and M. Hauge Dollerup, Towards representation stability for the second homology ofthe Torelli group, Geom. Topol. 16 (2012), no. 3, 1725–1765.[6] A. Borel, Density properties for certain subgroups of semi-simple groups without compact components,Ann. of Math. (2) 72 (1960), 179–188.[7] A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. ´Ecole Norm. Sup. (4) 7 (1974),235–272 (1975).[8] A. Borel, Stable real cohomology of arithmetic groups. II, in Manifolds and Lie groups (Notre Dame,Ind., 1980) , 21–55, Progr. Math., 14 Birkh¨auser, Boston, MA.[9] T. Brendle, D. Margalit and A. Putman, Generators for the hyperelliptic Torelli group and the kernelof the Burau representation at t “ ´
1, Invent. Math. (2015), no. 1, 263–310.[10] K. S. Brown,
Cohomology of groups , corrected reprint of the 1982 original, Graduate Texts in Math-ematics, 87, Springer, New York, 1994.[11] T. Church and B. Farb, Parameterized Abel-Jacobi maps and abelian cycles in the Torelli group, J.Topol. 5 (2012), no. 1, 15–38.[12] T. Church and B. Farb, Representation theory and homological stability, Adv. Math. 245 (2013),250–314.[13] T. Church and A. Putman, The codimension-one cohomology of SL n Z , to appear in Geom. Topol., arXiv:math/1507.06306 .[14] F. Cohen and J. Pakianathan, On automorphism groups of free groups and their nilpotent quotients,in preparation.[15] M. Day and A. Putman, A Birman exact sequence for Aut p F n q , Adv. Math. 231 (2012), no. 1, 243–275, arXiv:math/1104.2371 .[16] M. Day and A. Putman, The complex of partial bases for F n and finite generation of the Torellisubgroup of Aut p F n q , Geom. Dedicata 164 (2013), 139–153, arXiv:math/1012.1914 .[17] M. Day and A. Putman, A Birman exact sequence for the Torelli subgroup of Aut p F n q , Int. J. Alg.Comp. 26 (2016), no. 3, 585-617, arXiv:math/1507.08976 .[18] M. Day and A. Putman, A basis-free presentation for IA n , in preparation.
19] B. Farb, Automorphisms of F n which act trivially on homology, in preparation.[20] S. Galatius, Stable homology of automorphism groups of free groups, Ann. of Math. (2) 173 (2011),no. 2, 705–768.[21] A. Hatcher and K. Vogtmann, Cerf theory for graphs, J. London Math. Soc. (2) 58 (1998), no. 3,633–655.[22] A. Hatcher and N. Wahl, Stabilization for mapping class groups of 3-manifolds, Duke Math. J. 155(2010), no. 2, 205–269.[23] C. A. Jensen and N. Wahl, Automorphisms of free groups with boundaries, Algebr. Geom. Topol. 4(2004), 543–569.[24] D. Johnson, An abelian quotient of the mapping class group I g , Math. Ann. 249 (1980), no. 3,225–242.[25] N. Kawazumi, Cohomological aspects of Magnus expansions, preprint 2005.[26] S. Krsti´c and J. McCool, The non-finite presentability of IA p F q and GL p Z r t, t ´ sq , Invent. Math.129 (1997), no. 3, 595–606.[27] R. C. Lyndon and P. E. Schupp, Combinatorial group theory , Springer, Berlin, 1977.[28] W. Magnus, ¨Uber n -dimensionale Gittertransformationen, Acta Math. 64 (1935), no. 1, 353–367.[29] J. Nielsen, Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden, Math.Ann. 78 (1917), no. 1, 385–397.[30] A. Putman, An infinite presentation of the Torelli group, Geom. Funct. Anal. 19 (2009), no. 2, 591–643.[31] A. Putman, Obtaining presentations from group actions without making choices, Algebr. Geom. Topol.11 (2011), no. 3, 1737–1766.[32] A. Putman, The second rational homology group of the moduli space of curves with level structures,Adv. Math. 229 (2012), no. 2, 1205–1234.[33] T. Satoh, The abelianization of the congruence IA-automorphism group of a free group, Math. Proc.Cambridge Philos. Soc. 142 (2007), no. 2, 239–248.[34] T. Satoh, A survey of the Johnson homomorphisms of the automorphism groups of free groups andrelated topics, preprint 2012. Matthew DayDepartment of Mathematical Sciences, 301 SCENUniversity of ArkansasFayetteville, AR 72701E-mail: [email protected]
Andrew PutmanDepartment of MathematicsRice University, MS 1366100 Main St.Houston, TX 77005E-mail: [email protected]@math.rice.edu