Arithmetic group symmetry and finiteness properties of Torelli groups
Abstract.
We examine groups whose resonance varieties, characteristic varieties and Sigmainvariants have a natural arithmetic group symmetry, and we explore implications on various finiteness properties of subgroups. We compute resonance varieties, characteristic varieties and Alexander polynomials of Torelli groups, and we show that all subgroups containing the Johnson kernel have finite first Betti number, when the genus is at least . We also prove that, in this range, the adic completion of the Alexander invariant is finitedimensional, and the Kahler property for the Torelli group implies the finite generation of the Johnson kernel.
Key words and phrases:
resonance variety, characteristic variety, Alexander invariant, Alexander polynomial, Sigmainvariant, arithmetic group, semisimple Lie algebra, finiteness properties, Torelli group, Johnson kernel, Kahler group2000 Mathematics Subject Classification:
Primary 20J05, 57N05; Secondary 11F23, 20G05.Contents
1. Introduction and statement of results
The natural symmetry factors through , for a good number of invariants of a group . The simplest examples are the abelianization, , and the quotient of by its torsion, . More wellknown examples are provided by the cohomology ring and the graded Lie algebra associated to the lower central series, .
Our starting point in this work is to consider a group epimorphism, , with finitely generated kernel , and to examine three other known types of invariants with natural outer symmetry, through the prism of the symmetry induced by the canonical homomorphism, . Firstly, we look at the resonance varieties (i.e., the jump loci for a certain kind of homology, associated to the ring ), sitting inside ; they are reviewed in Section 3, and their outer symmetry is discussed in Remark 3.1. Secondly, we inspect the characteristic varieties (i.e., the jump loci for homology with rank one complex local systems), lying inside the character torus , and their intersection with ; their definition is recalled in Section 5, and their outer symmetry is explained in Lemma 5.1. Finally, we recollect in Section 6 a couple of relevant facts about the BieriNeumannStrebelRenz (BNSR) invariants, , and we point out their outer symmetry in Lemma 6.1.
Our choice for the types of invariants was dictated by the fact that each of them controls a certain kind of finiteness properties. We begin with resonance, for which this relationship seems to be new. We devote Section 4 to the analysis of the connexions between triviality of resonance and finiteness of various (completed) Alexandertype invariants, for finitely generated groups.
Our motivation comes from the Torelli groups, , consisting of the isotopy classes of homeomorphisms of a closed, oriented, genus surface, inducing the identity on the first homology group of the surface. Their finite generation was proved by D. Johnson [21], for . According to Hain [16], is a formal group in the sense of D. Sullivan [36], for , but is not formal.
Note also that the full mapping class group is an extension of the integral symplectic group by , for :
(1.1) 
In particular, the resonance and characteristic varieties of , as well as its BNSRinvariants, acquire a natural symmetry, for .
Let be a subgroup containing the derived group . Then becomes in a natural way a module over the group ring , with module structure induced by conjugation. Its adic completion is denoted , where is the augmentation ideal. When , is the classical Alexander invariant from link theory (over ). The technique of adic completion was promoted in lowdimensional topology by Massey [28]. Here is our first main result.
Theorem A.
Let be a finitely generated group.

Assume is formal. Then if and only if .

Let be a subgroup containing . Then , if .
Aiming at finer finiteness properties, we go on by examining characteristic varieties, in Section 5. Here, we start from a basic result of Dwyer and Fried [14], as refined in [31]. It says that the finiteness of Betti numbers, up to degree , of normal subgroups with free abelian quotient is detected precisely by the finiteness of the intersection between characteristic varieties of type , for , and the corresponding connected subtorus of .
Given the symmetry of characteristic varieties, we are thus led to consider the following context, inspired by Torelli groups. Let be a module which is finitely generated and free as an abelian group. Assume that is an arithmetic subgroup of a simple linear algebraic group defined over , with . Suppose also that the action on extends to an irreducible, rational representation in . (Note that the above assumptions are satisfied for by and , due to Johnson’s pioneering results on the symplectic symmetry of Torelli groups from [20, 23].) The representation in gives rise to a natural action on the connected affine torus .
Theorem B.
If the module satisfies the above assumptions, then is geometrically irreducible, that is, the only invariant, Zariski closed subsets of are either equal to , or finite.
We deduce Theorem B from a deep result in diophantine geometry, due to M. Laurent [25], in Section 5. Note that the conclusion of our theorem above is in marked contrast with the behavior of the induced representation in the affine space , for which invariant, infinite and proper Zariski closed subsets may well exist. Theorem B enables us to obtain in Section 5 the following consequences of the triviality of resonance.
Theorem C.
Let be a group epimorphism with finitely generated kernel , having the property that . Assume is arithmetic, where the linear algebraic group is defined over , simple, with . Suppose moreover that the canonical representation in extends to an irreducible, rational representation in . The following hold.

The intersection is finite, for all .

If moreover , the Alexander polynomial is a nonzero constant, modulo the units of the group ring .

For any subgroup , containing the kernel of the canonical map, , the first Betti number is finite.
Note that the computation of characteristic varieties and Alexander polynomials can be a very difficult task, in general. What makes life easier in Theorem C, Parts (1) and (2), is the arithmetic symmetry. As far as Torelli groups are concerned, we obtain the following results. Denote by the kernel of the canonical map, , identified by Johnson in geometric terms in [22].
Theorem D.
Let be the Torelli group in genus . The following hold.

The resonance (in degree and depth ) is trivial: .

The (reduced) characteristic varieties are finite, for all .

The Alexander polynomial is a nonzero constant, modulo units.

For any subgroup containing the Johnson kernel , the vector space is finitedimensional.

For any subgroup containing , .
We compute resonance varieties of Torelli groups in Section 3. Note that is nontrivial; this is an easy consequence of a basic result of Hain [16], who computed the graded Lie algebra , truncated up to degree , for . His result, formulated in the category of representations, is recorded in Section 2. We obtain the triviality of , for , by exploiting the symmetry. Parts (2)(3) of Theorem D follow from Theorem C.
Parts (4)(5) of Theorem D appear to be rather surprising, since the general belief in the literature seems to be that Torelli groups and other related groups should exhibit infiniteness properties; see for instance [15, §5.1] and the survey [17]. Theorem D(4) is a consequence of Theorem C(3), and Theorem D(5) follows from Theorem A(2). Note that the finitedimensionality of the (uncompleted) Alexander invariant cannot be deduced from Theorem D(2), since contains nontrivial torsion, according to Johnson [23]. To the best of our knowledge, the finiteness of is an open question.
Theorem D(4) solves two of the most popular finiteness problems related to Torelli groups, as recorded in Farb’s list from [15]: for , we answer Question 5.2 (over ); the general case () settles Problem 5.3 (at the level of first Betti numbers).
According to [6, erratum], the finite generation of is an open question, for . Another important problem concerns the Kahler property: is a Kahler group (that is, the fundamental group of a compact Kahler manifold)? The answer is negative, in low genus: is not finitely generated, as proved by McCullough and Miller [29], and violates the formality property of Kahler groups, established by Deligne, Griffiths, Morgan and Sullivan in [10]. Theorem 4.9 from [15] states that is not a Kahler group for . Since the proof assumed infinite generation of , the Kahler group problem is open as well.
We investigate in Section 6 the BNSR invariants, which control the finiteness properties of normal subgroups with abelian quotient [4, 5]. We use their arithmetic symmetry, together with Delzant’s description from [11] of the first BNSR invariant of Kahler groups, and our Theorem D(1), to prove the next result. It connects the Kahler problem for Torelli groups with the finite generation question for Johnson kernels. See Remark 6.6 for more details on our strategy of proof.
Theorem E.
If is a Kahler group, the Johnson kernel must be finitely generated, for .
2. Associated graded Lie algebra
The symplectic symmetry is wellknown to be an important tool for the study of Torelli groups. In this section, we review a basic result from Hain’s work [16], related to the symplectic symmetry at the level of the associated graded Lie algebra.
Let be a closed, oriented, genus surface. Denote by the associated mapping class group, i.e., the group of isotopy classes of orientationpreserving homeomorphisms of . Let be the group of symplectic automorphisms of . The arithmetic group is a Zariski dense subgroup of the semisimple algebraic group ; see e.g. [33, Corollary 5.16]. The Torelli group is the kernel of the natural homomorphism , which associates to an element of the induced action on . The defining exact sequence of the group is
(2.1) 
R. Hain’s starting point is D. Johnson’s pioneering work from [20, 21, 22, 23], which we review first. In the sequel we will assume that . In this range, Johnson proved [21] that the group is finitely generated. For a group , we denote by the quotient of its abelianization by the torsion subgroup. Among other things, Johnson [20, 23] gave a very convenient description of , in the following way.
Fix a symplectic basis of , , and denote by the symplectic form. Note that the action on canonically extends to a action on the exterior algebra , by graded algebra automorphisms. Consider the equivariant embedding, , given by , and denote by the module . Johnson’s homomorphism, constructed in [20], will be denoted by .
Theorem 2.1 (Johnson).
The group homomorphism is equivariant, with respect to the (left) conjugation action on induced by (2.1), and the restriction of the action on via . It induces a equivariant isomorphism, .
Setting and , it follows that the canonical representation of , coming from (2.1), in , extends to a rational representation of . This symplectic symmetry propagates to higher degrees, in the following sense.
Recall that the associated graded Lie algebra (with respect to the lower central series) of a group , , is generated as a Lie algebra by .
Lemma 2.2.
Given a group extension,
(2.2) 
assume that is finitely generated, is a Zariski dense subgroup of a complex linear algebraic group , and the action on extends to a rational representation of in . Then the action on extends to an action of on , in the category of graded rational representations; moreover, every acts on by a graded Lie algebra automorphism.
Proof.
Presumably this result is wellknown to the experts. Being unable to find a reference, we decided to include a proof. Set , noting that acts on by graded Lie algebra automorphisms. For each , denote by the kernel of the linear surjection sending to . Since is Zariski dense, the linear subspace is invariant. It remains to show that the rational representations of in constructed in this way, which extend the action coming from (2.2), have the property that , for , and . This in turn is easily proved by induction on . Induction starts with , by noting that the iterated Lie bracket, , is equivariant by construction. For the inductive step, write , with and , and use the Jacobi identity to conclude. ∎
By Lemma 2.2 and Theorem 2.1, we have a short exact sequence of rational representations,
(2.3) 
where denotes the Lie bracket. Hain [16] computed the associated exact sequence of modules, where is the Lie algebra of . To describe his result, we follow the conventions from [16, Section 6]. Our references for algebraic groups (respectively Lie algebras) are [19] (respectively [18]).
The Lie algebra of the maximal diagonal torus in is denoted , and has coordinates . Let be the corresponding root system, with the standard choice of positive roots, . Let be the canonical decomposition of the Lie algebra. We denote by the associated Borel subgroup of , with unipotent radical ; the Lie algebra of is , and the Lie algebra of is . We work with the finitedimensional modules associated to rational representations of . The irreducible ones are of the form , where the dominant weight is a positive integral linear combination of the fundamental weights, .
It follows from Theorem 2.1 that , as modules. According to [16, Lemma 10.2], all irreducible submodules of occur with multiplicity one, and contains as a submodule.
Theorem 2.3 (Hain).
The map from (2.3) is the canonical equivariant projection of onto the submodule .
3. Resonance varieties of Torelli groups
In this section, we use representation theory to compute the resonance varieties (in degree and for depth ) of the Torelli groups , for , over .
We begin by reviewing the resonance varieties, , associated to a connected, gradedcommutative algebra , for (degree) and (depth) . Given , denote by leftmultiplication by in , noting that , due to gradedcommutativity. Set
(3.1) 
Remark 3.1.
It seems worth pointing out that, given an arbitrary group extension (2.2), one has the following symmetry property, related to resonance. By standard homological algebra (see e.g. [8]), the conjugation action of on induces a (right) action of on , by graded algebra automorphisms. We conclude from (3.1) that the representation on leaves invariant, for all and .
In this paper, we need to consider only the resonance varieties , which plainly depend only on the corestriction of the multiplication map, , to its image. When , it is easy to see that each is a Zariski closed, homogeneous subvariety of the affine space . For a connected space , we denote by . For a group , means . By considering a classifying map, it is immediate to check that . We will abbreviate by .
There is a wellknown, useful relation between cupproduct in low degrees and group commutator, see Sullivan [35], and also Lambe [24] for details. For a group with finite first Betti number, there is a short exact sequence,
(3.2) 
where cohomology is taken with coefficients and is dual to the Lie bracket from (2.3). (Note that the only finiteness assumption needed in the proof from [24] is .)
In what follows, we retain the notation from Section 2. Set and . We infer from (2.3) and (3.2) that is invariant. Here is our first step in exploiting the complex symmetry from Theorem 2.3.
Lemma 3.2.
If , then contains a maximal vector of the module .
Proof.
This is an easy consequence of Borel’s fixed point theorem [19, Theorem 21.2], which guarantees the existence of a invariant line, . Invariance under the action of the maximal torus implies that belongs to a weight space of the action on , that is, for some . Finally, follows from invariance. ∎
Since belongs to the Weyl group of , all finitedimensional representations of are selfdual [18, Exercise 6 on p.116]. This remark leads to the following dual reformulation of Theorem 2.3, via (3.2).
Lemma 3.3.
Set . Then , as modules, and the kernel of the cupproduct, , is .
Theorem 3.4.
For , , while , for .
The assertion for is an immediate consequence of Lemma 3.3, since in this case , cf. Lemma 10.2 from [16]. So, we will assume in the sequel that and , and use Lemma 3.2 to derive a contradiction.
Set . We will need explicit maximal vectors, for and for . To this end, we recall that , where the action on is by algebra derivations; in particular, , for and . Set , and . Denote by the class of in , and let be the class of in . Using the explicit description of from [16, Section 6], it is straightforward to check that has weight , has weight , and .
To verify that both and are nonzero, we will resort to the equivariant contraction constructed by Johnson in [20, p.235], , given by
(3.3) 
where the dot designates the intersection form on . Set , and . It follows from [20, pp.238239] that there is an induced isomorphism,
(3.4) 
Clearly, and . Hence, both and are nonzero, as needed.
We infer from uniqueness of maximal vectors [18, Corollary 20.2] that necessarily , if ; see Lemma 3.2. Denote by the submodule of from Lemma 3.3. By definition (3.1) and Lemma 3.3, if and only if , where denotes leftmultiplication by in the exterior algebra.
Since , we conclude that is equivariant. Since , it follows that
(3.5) 
for each weight subspace .
Lemma 3.5.
If , then contains a nonzero vector killed by .
Proof.
Taking into account the preceding discussion, the assertion will follow from Engel’s theorem [18, Theorem 3.3], applied to the Lie algebra and the module . It is enough to check that the action of any on is nilpotent. This in turn follows from the fact that , where , for any and any nontrivial weight space . Indeed, for big enough. To check the vanishing claim, one may invoke [18, Theorem 20.2(b)], and then use a height argument. ∎
Lemma 3.6.
The following hold.

For , the vector is nonzero.

The class of in does not belong to .
Proof.
We will finish the proof of Theorem 3.4, by showing that Lemmas 3.5 and 3.6 lead to a contradiction, assuming and .
If , then , by uniqueness of maximal vectors. Taking the weight decomposition of , it follows from (3.5) that we may suppose that in (3.6). Therefore, , contradicting Lemma 3.6(1).
If , then , for some , by the same argument on weight decomposition as before. Since the weights and are conjugate under the action of the Weyl group, the weight space is onedimensional, generated by the class of in ; see [18, Theorems 20.2(c) and 21.2], [16, Section 6] and (3.4). Therefore, , contradicting Lemma 3.6(2). The proof of Theorem 3.4 is thus completed.
4. Resonance and finiteness properties
We devote this section to a general discussion of finiteness properties related to resonance, with an application to Torelli groups. Unless otherwise mentioned, we work with coefficients. For a graded object , the notation means that we forget the grading.
We need to review a couple of key notions. We begin with the holonomy Lie algebra associated to a connected CWcomplex with finite skeleton, . This is a quadratic graded Lie algebra, defined as the quotient of the free Lie algebra generated by , , graded by bracket length, modulo the ideal generated by the image of the comultiplication map, ; here we are using the standard identification , given by the Lie bracket. Note that the dual of is the cupproduct, .
When is a finitely generated group and , is denoted . By considering a classifying map, it is easy to see that , for a connected CWcomplex with finite skeleton. The associated graded Lie algebra , denoted , is also finitely generated in degree one, but not necessarily quadratic. The canonical identification, , extends to a graded Lie algebra epimorphism, . By (3.2), this factors to a graded Lie algebra surjection,
(4.1) 
Let be the polynomial algebra , endowed with the usual grading. For a Lie algebra , denote by the derived Lie algebra, and by the second derived Lie algebra. The exact sequence of graded Lie algebras
(4.2) 
yields a positively graded module structure on the infinitesimal Alexander invariant, , induced by the adjoint action. A finite presentation of is described in [30, Theorem 6.2].
Let be the elementary ideal generated by the codimension minors of a finite presentation for . Denote by the zero set of . The next lemma explains the relationship between the infinitesimal Alexander invariant and the resonance varieties in degree one.
Lemma 4.1.
Let be a finitely generated group. Then the equality
holds for all .
Proof.
We know from [30, Theorem 6.2] that , as modules, where
(4.3) 
and the linear map is given by , for .
Pick any . By linear algebra, we infer that if and only if . Consider the exact cochain complex , where denotes left multiplication by , and the dual exact chain complex, . It is straightforward to check that the restriction of to equals , where is as in (4.3).
Denoting by the restriction of to , we obtain from exactness the following isomorphism:
By exactness again, , where . Hence if and only if .
Plainly, the linear map dual to is . Consequently, if and only if , that is, if and only if ; see definition (3.1). ∎
We may spell out our first general result relating resonance and finiteness.
Theorem 4.2.
Let be a finitely generated group. Then if and only if .
Proof.
Lemma 4.1 yields in particular the equality , away from the origin. Let be the maximal ideal of .
If , then , for some , by degree inspection. Taking zero sets, we obtain that .
Conversely, the assumption implies , by Hilbert’s Nullstellensatz. Therefore, , for some . Since is finitely generated over , we infer that . ∎
Theorem 4.2 has several interesting consequences. To describe them, we first recall a couple of notions from rational homotopy theory. A Malcev Lie algebra (in the sense of Quillen) is a Lie algebra endowed with a decreasing vector space filtration satisfying certain axioms; see [32, Appendix A], where Quillen associates in a functorial way a Malcev Lie algebra, , to an arbitrary group . Following Sullivan [36], we say that a finitely generated group is formal if is the completion with respect to degree of a quadratic Lie algebra, as a filtered Lie algebra. In Theorem 1.1 from [16], Hain proves that the Torelli group is formal, for (but is not formal). Fundamental groups of compact Kahler manifolds are formal, as shown by Deligne, Griffiths, Morgan and Sullivan in [10]. Many other interesting examples of formal groups are known; see e.g. [13] and the references therein.
Next, we review the Alexander invariant, , associated to a finitely generated group . The exact sequence of groups
(4.4) 
may be used to put a finitely generated module structure on , induced by conjugation, over the Noetherian group ring . We denote by the augmentation ideal, and by the adic completion of a module .
Given a graded vector space , means completion with respect to the degree filtration: , where . The canonical filtration of is , where is the th projection of the inverse limit. The next corollary proves Theorem A(1).
Corollary 4.3.
Let be a finitely generated, formal group. Then if and only if . In particular, , when .
Proof.
Let be the maximal ideal of the origin. The formality of provides a vector space isomorphism between and the adic completion of , cf. Theorem 5.6 from [13]. Since is generated in degree , its adic completion coincides with the degree completion, . Clearly, and are simultaneously finitedimensional. Hence, our first claim follows from Theorem 4.2. For the second claim, simply note that the completion of a finitedimensional vector space is again finitedimensional. ∎
Remark 4.4.
The finitely generated group discussed in Example 6.4 from [31] satisfies , yet . It can be shown that is formal. Consequently, is only a necessary condition for the finitedimensionality of , in general.
In the particular case when is nilpotent, the condition is automatically satisfied, since is finitely generated. We recover in this way from Corollary 4.3 the resonance obstruction to formality of finitely generated nilpotent groups found by Carlson and Toledo in [9, Lemma 2.4]. We refer the reader to Macinic [27], for similar higherdegree obstructions to the formality of a finitely generated nilpotent group.
As we shall see below, the main point in Corollary 4.3 is the fact that the finitedimensionality of forces , when is formal. We point out that formality is needed for this implication. Indeed, let be the finitely generated nilpotent Heisenberg group with . As noted before, . Yet, the resonance variety is nontrivial; see for instance [27, Proposition 5.5].
Let be an arbitrary finitely generated group, and a subgroup containing . Clearly, is normal in , and conjugation makes a finitely generated module over the Noetherian group ring . By restriction via the canonical epimorphism, , becomes a finitely generated module. We may now state our next main result from this section, which proves Theorem A(2).
Theorem 4.5.
Let be a finitely generated group, and a subgroup containing . If , the vector space is finitedimensional.
Proof.
We first treat the particular case , where we know from Theorem 4.2 that the vector space is finitedimensional. The canonical graded Lie algebra surjection, , composed with the epimorphism (4.1), gives a graded Lie algebra surjection, , that factors through an epimorphism
(4.5) 
It follows from (4.5) that , for .
On the other hand, we have an isomorphism,
(4.6) 
for ; see [28, pp.400–401]. We infer from (4.6) that the adic filtration of stabilizes, for . Therefore, , as asserted.
For the general case, consider the extension
(4.7) 
where is a finitely generated subgroup of , and denote by the coinvariants of the module , noting that the canonical projection, , is linear.
The HochschildSerre spectral sequence of (4.7) with trivial coefficients (see e.g. [8, p.171]) provides an exact sequence of finitely generated modules,
(4.8) 
By standard commutative algebra (see for instance [2, Chapter 10]), the adic completion of (4.8) is again exact. Since , our claim about follows from the finitedimensionality of . ∎
Corollary 4.6.
Let be a subgroup of containing . The adic completion of is finitedimensional, for , where is the augmentation ideal of the group ring.
5. Characteristic varieties of Torelli groups
Guided by the interplay between arithmetic and Torelli groups, coming from (2.1), we will examine now groups whose characteristic varieties possess a natural discrete symmetry. We will compute the (restricted) characteristic varieties of , in degree , when , and deduce that .
Our setup in this section is the following. Let
(5.1) 
be a group extension, where is finitely generated and is an arithmetic subgroup of a complex linear algebraic group , defined over , simple and with rank at least . The motivating examples are the extensions (2.1), for . Under the above assumptions on , we recall that any finite index subgroup is Zariski dense in , as follows from Borel’s density theorem; see e.g. [33, Corollary 5.16].
We continue by reviewing a couple of relevant facts related to character tori and characteristic varieties. Let be a finitely generated group. The character torus is a linear algebraic group, with coordinate ring the group algebra . The connected component of is .
For the beginning, we need to assume in (5.1) only that is finitely generated. The natural representation in (respectively ) induced by conjugation canonically extends to and (respectively to and ). The corresponding left action on , by algebraic group automorphisms, is denoted by , for and , and is defined by , for