Algebraic Completions of Discrete and Profinite Groups

Motivation Unipotent Completion Relative Unipotent Completion The Proﬁnite Case Weighted Completion

Algebraic Completions of Discrete and Proﬁnite Groups

Richard Hain

Duke University

November 6, 2008

Richard Hain Algebraic Completions of Discrete and Proﬁnite Groups Motivation Unipotent Completion What? Relative Unipotent Completion Why? The Proﬁnite Case How? Weighted Completion The goal

The problem is to “linearlize” a given a discrete (or proﬁnite) group Γ. That is, we want to turn Γ into a vector space over a given ﬁeld F. Naive Example ab We can replace Γ by H1(Γ, F) := Γ ⊗Z F. This is not very enlightening as, in general, we lose too much information — such as all information about the non-abelian properties of Γ.

Richard Hain Algebraic Completions of Discrete and Proﬁnite Groups Motivation Unipotent Completion What? Relative Unipotent Completion Why? The Proﬁnite Case How? Weighted Completion Why?

1 To understand Γ by “simplifying” it. In general, ﬁnitely presented groups are difﬁcult to understand, even if you have a presentation. If you don’t believe this, I will give you a nice presentation of the mapping class group Γg in any genus ≥ 2, and you can tell 2 me what H (Γg, V ) is, where V is any non-trivial representation of Spg, pulled back along the natural homomorphism Γg → Spg. 2 To accomodate and study “motivic” structures on Γ, such as Hodge structures, (linear) Galois actions, etc.

Richard Hain Algebraic Completions of Discrete and Proﬁnite Groups Motivation Unipotent Completion What? Relative Unipotent Completion Why? The Proﬁnite Case How? Weighted Completion How?

We will replace Γ by an afﬁne proalgebraic group G over F. An afﬁne proalgebraic group is just an inverse limit

G = ←−lim Gα α

of afﬁne F-groups G. Knowing G is equivalent to knowning its coordinate ring O(G) := O( ), −→lim Gα α which is a Hopf algebra — and so a vector space. G g := g We can also replace by its Lie algebra −→lim α.

Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Constructions Unipotent Completion Examples Relative Unipotent Completion Presentations The Profinite Case More structure Weighted Completion More Examples Definition

Unipotent completion, also called Malcev completion, has its origins in a theorem of Malcev and is probably due to Quillen (1969). Fix a ﬁeld F of characteristic zero, such as Q. The unipotent completion of Γ over F is deﬁned by

Γun = Γun := lim U /F ←− ρ ρ

where Uρ is a unipotent F-group and ρ :Γ → Uρ(F) is a Zariski dense homomorphism. There is a natural homomorphism ρ˜ :Γ → Γun(F). Every homomorphism from Γ to a unipotent F-group factors through ρ˜. Consequently, the category of unipotent representations of Γ over F is the category of ﬁnite dimensional representations of Γun.

Let J be the kernel of the augmentation : FΓ → F. The J-adic completion of FΓ is

Γ∧ : Γ/ n. F ←−lim F J n This is a complete Hopf algebra. Then

Γun(F) = group-like elets of FΓ∧ = {x ∈ 1 + J∧ : ∆x = x ⊗ x}.

Its Lie algebra is the set of primitive elements

{x ∈ FΓ∧ : ∆(x) = 1 ⊗ x + x ⊗ 1}

of FΓ∧ and is pronilpotent.

1 The completion of a ﬁnitely generated abelian group Γ is n un n Γ ⊗Z F. For example (Z ) = (Ga) : 1 n 1 F n n =∼ Z → =∼ (F)n. Z 0 1 0 1 Ga

2 The unipotent completion of the integral Heisenberg group Γ is the Heisenberg group H:     1 ZZ 1 FF Γ = 0 1 Z → 0 1 F = H(F). 0 0 1 0 0 1

If Γ is the free group Γ = hx1,..., xni, then

∧ ' θ : FΓ −→ FhhX1,..., Xnii

is a (complete) Hopf algebra isomorphism where each Xj is deﬁned to be primitive and

θ(xj ) = exp Xj j = 1,..., n.

un ' ∧ θ induces an isomorphism Γ (F) −→ exp L(X1,..., Xn) . The un ∧ Lie algebra of Γ is L(X1,..., Xn) .

un If Γ = hx1,..., xn : r1,..., rmi, then the Lie algebra of Γ is

∧ L(X1,..., Xn) /(log θ(rj ): j = 1,..., m).

This works in simple cases, but is not useful in general. Examples

1 If H1(Γ, F) = F (e.g., Γ is a knot group or a braid group Bn), un ∼ then Γ = Ga. 2 If H1(Γ, F) = 0 (e.g. Γ is a mapping class group or an irreducible lattice in a semi-simple Lie group of real rank ≥ 2), then Γun is trivial.

These illustrate some limitations of unipotent completion.

Suppose that u is a pronilpotent Lie algebra over F. A section s of u → H1(u) induces a continuous surjection

∧ f := L(H1(u)) → u,

so that u =∼ f/r where r ⊆ [f, f]. The image of any section of

r → r/[r, f] = H2(u)

will be a minimal set of relations. Thus u has a minimal presentation of the form

∧ u = L(H1(u)) /(H2(u))

∧ ∧ for a suitable embedding H2(u) ,→ [L , L ]. Richard Hain Algebraic Completions of Discrete and Proﬁnite Groups Motivation Constructions Unipotent Completion Examples Relative Unipotent Completion Presentations The Proﬁnite Case More structure Weighted Completion More Examples Basic structure theorem

un The Lie algebra u of Γ is controlled by H1(u) and H2(u). Theorem There are natural homomorphisms

Hk (Γ, F) → Hk (u) ∼ If Γ is a ﬁnitely presented group, then H1(u) = H1(Γ, F) and H2(Γ, F) → H2(u) is surjective. The homomorphism

∧ ∧ ∧ ∧ ∧ ∼ 2 H2(u) → [L , L ]/[L , L , L ] = Λ H1(u)

is dual to the cup product Λ2H1(u) → H2(u).

There is also a version for Γ that are not ﬁnitely presented.

Theorem If X is an algebraic variety over a subﬁeld K of C, then un 1 (Morgan, Hain) π1(X( ), x) has a natural mixed Hodge C /Q structure for all x ∈ X(C); un 2 (Deligne) π1(X( ), x) has a natural GK -action for all C /Q` K -rational points x of X.

Corollary (Morgan (1978)) If X is smooth complex algebraic variety, then the Lie algebra of un π1(X, x) has a minimal presentation with generators of weights in {−1, −2} and relations of weights in {−2, −3, −4}.

In the 1960’s Serre asked if every ﬁnitely presented group could occur at the fundamental group of a smooth quasi projective variety X over C. un Morgan’s result implies that the Lie algebra u of π1(X, x) is “weighted homogeneous” with at at worst quadratic, cubic and quartic relations.

Example (Morgan) The group Γ = hx, y :[x, [x, [x, [x, y]]]]i is not the fundamental group of any smooth quasi-projective variety.

Example (Kohno)

The pure braid group Pn is the fundamental group of a hyperplane complement Xn whose H1 is pure of weight −2. un This implies that the Lie algebra of Pn is quadratically presented:

∧ pn = L(Xjk ) /([Xij , Xkm], [Xij , Xik + Xjk ]: i, j, k, m distinct)

2 1 2 The quadratic relations are dual to Λ H (Xn) → H (Xn); they are easily computed from Brieskorn’s description of the cohomology ring of Xn.

Similarly, the Lie algebra of the unipotent completion of π1 of any compact Kähler manifold X is quadratically presented.

Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Generalities Unipotent Completion Structure Relative Unipotent Completion Examples The Profinite Case More structure Weighted Completion Definition

Suppose that Γ is a discrete group, S is a reductive F-group and that ρ :Γ → S(F) is a Zariski dense representation. The completion of Γ relative to ρ is

G := ←−lim Gφ φ

where φ ranges over all Zariski dense representations Γ → Gφ(F) such that: (1) Gφ is an afﬁne F group that is an extension of S by a unipotent group:

1 → Uφ → Gφ → S → 1

and; (2) the composite Γ → Gφ(F) → S(F) is ρ.

Unipotent completion is the case where S is trivial. Note that G is an extension

1 → U → G → S → 1

U = of S by the prounipotent group ←−lim Uφ. There is a natural Zariski dense homomorphism

ρ˜ :Γ → G(F).

Every homomorphism of Γ to an F-group G that is an extension of S by a unipotent group, and where Γ → G(F) → S(F) is ρ, factors through ρ˜.

If Γ satisfies some mild finiteness conditions (e.g., Γ is finitely presented), then Levi’s Theorem implies that G → S splits, so that ∼ G = S n exp u where u is a pronilpotent Lie algebra in the category of S-modules. Thus, to describe G, we need to find a presentation of u as a Lie algebra in the category of S-modules. For simplicity, we’ll suppose that all irreducible representations of S are absolutely irreducible.

Let {Vλ}λ be a set of representatives of the isomorphism classes of irreducible S-modules. Theorem If Γ is ﬁnitely presented and all S-modules are absolutely irreducible, then

∼ Y 1 ∗ H1(u) = H (Γ, Vλ) ⊗ Vλ λ∈Sˇ

as S-modules, and there is a natural S-equivariant surjection:

Y 2 ∗ H (Γ, Vλ) ⊗ Vλ → H2(u) λ∈Sˇ

Take F = C, Γ = Bn to be the full braid group, S to be the symmetric group Sn, and ρ to be the standard homomorphism Bn → Sn. Note that Sn acts on pn by σ : Xjk → Xσ(j)σ(k). Then

G = Sn n exp pn.

The homomorphism Bn → G(C) is given by integrating the KZ-equation.

If Γ is an irreducible lattice in a semi-simple real Lie group S of real rank ≥ 2 (e.g., Γ = SLn(Z), n ≥ 3), then the completion of Γ with respect to the inclusion Γ ,→ S(R) is S. This follows from Raghunathan’s vanishing result:

H1(Γ, V ) = 0

for all S-modules V , which implies that the prounipotent radical U is trivial.

Suppose that X is a compact oriented surface of genus g. Take Γ to be the mapping class group of X:

+ Γg := π0 Diff X.

Take F = Q and S = Spg. The representation ρ :Γg → Spg(Q) is given by the action of Γg on H1(X). This is Zariski dense as its image is Spg(Z).

The Torelli group Tg is the kernel of ρ:

1 → Tg → Γg → Spg(Z) → 1.

Denote the completion of Γg with respect to ρ by Gg and its prounipotent radical by Ug. Since Tg → Ug(F), there is a un un homomorphism Tg → Ug. Denote the Lie algebra of Tg by tg. Theorem (Hain, 1997) If g ≥ 3, then there is an exact sequence

un 0 → Ga → Tg → Gg → Spg → 1.

un The kernel of T2 → U2 is very large. The theorem also fails in genus 1 as we will soon see.

Inﬁnitesimal presentation of Tg

Although Tg is not known to be ﬁnitely presented for any g ≥ 3, its unipotent completion is. Theorem (Hain, 1997)

If g ≥ 3, then tg is ﬁnitely presented with at worst cubic relations. If g ≥ 6, then ∼ ∧ tg = L V[13] /(R)

where R is the unique Spg-invariant complement of V[2,2] ⊕ Q in 2 Λ V[13]. In particular, tg is ﬁnitely presented for all g ≥ 3.

This uses Dennis Johnson’s work on the Torelli group and Kabanov’s Purity Theorem in an essential way.

For simplicity, take Γ to be the full modular group SL2(Z). Take S to be SL2 and ρ :Γ → S(Q) to be the inclusion ρ : SL2(Z) ,→ SL2(Q). Denote the completion of SL2(Z) with respect to ρ by G. Question What is the prounipotent radical U of G?

2 n Since Γ is virtually free, H (SL2(Z), S H) = 0 for all n ≥ 0. This ∧ implies that u is free, so that u = L(H1(u)) . Note that

Y 1 n ∗ n H1(u) = H (SL2(Z), S H) ⊗ S H. n≥0

The computation of H1(u) is provided by: Theorem (Eichler-Shimura) For all n ≥ 0,

1 n H (SLn(Z), S H) = Mn+2 ⊕ Mn+2 ⊕ CGn+2

where Mk denotes the space of cusp forms of SL2(Z) of weight k and Gk denotes the Eisenstein series of weight k. These all vanish when k is odd. . . . and the classical fact: ( bm/6c − 1 m ≡ 1 mod 6 dim M2m = bm/6c m 6≡ 1 mod 6.

If Γ is the (geometric) fundamental group of smooth variety deﬁned (say) over a subﬁeld of C, and if ρ is the monodromy representation of a suitable local system (a polarized variation of Hodge in the Hodge case, and a lisse sheaf in the Galois case), then O(G) (resp. g) will be an ind-object (resp. pro-object) of the category of mixed Hodge structures in the Hodge case (Hain, 1997) or GF -modules in the Galois case (Hain-Matsumoto, 2003). This additional structure was used in an essential way to compute the presentation of tg.

Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Unipotent Completion Profinite groups Relative Unipotent Completion Continuous relative completion The Profinite Case Weighted Completion Profinite groups

Recall that a proﬁnite group is an inverse limit of ﬁnite groups. Examples

1 The Galois group of a ﬁeld K with algebraic closure K :

:= ( / ) GK ←−lim Gal L K [L:K ]<∞, L⊆K

2 The proﬁnite completion of a discrete group Γ:

Γ∧ := Γ/ ←−lim N N/Γ, |Γ:N|<∞

They are totally disconnected compact topological groups.

Suppose that Γ is profinite. Take F to be the topological field Q` (for a fixed prime number `) and ρ :Γ → S(Q`) be a continuous, Zariski dense homomorphism, where S(Q`) is regarded as a topological group in the `-adic topology. The continuous completion of Γ relative to ρ is the inverse limit of the Gφ over all continuous Zariski dense homomorphisms φ :Γ → Gφ(Q`) that lift ρ, where Gφ is an extension of S by a unipotent group. Question How does the completion of a finitely generated discrete group Γ with respect to ρ :Γ → S(Q) compare to the continuous completion of its profinite completion Γ∧ with respect to ∧ ∧ ρ :Γ → S(Q`)?

Denote the completion of the discrete group Γ with respect to ρ by G, where F = Q. Theorem (Hain-Matsumoto) The canonical homomorphism ρ˜ :Γ → G(Q) extends to a ∧ ∧ ∧ homomorphism ρ˜ :Γ → G(Q`) and (G ⊗Q Q`, ρ˜ ) is the continuous completion of Γ∧ relative to ρ∧.

Corollary (Sample) For each smooth projective curve C of genus g deﬁned over Q, there is a natural action of G on Gg, the completion of ∼ Q π1(Mg, [C]) = Γg with respect to Γg → Spg(Q`).

Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Definition Unipotent Completion Presentation Relative Unipotent Completion Completions of Galois groups The Profinite Case Completion of the arithmetic modular group Weighted Completion Preliminaries

Suppose that we have a central cocharacter χ : Gm → S. Examples

1 −1 S = GSpg and χ : Gm → GSpg takes z to z id −2 2 S = Gm and χ : Gm → Gm takes z to z (g = 0 case!) Schur’s Lemma implies that for each irreducible S-module V , there is an integer nV such that Gm acts on V (via χ) with weight nV . Deﬁnition

An irreducible S-module V is negatively weighted if nV < 0. An S-module is negatively weighted if all of its irreducible factors are.

Take the base field F to be Q`. Suppose that Γ is a profinite group and that ρ :Γ → S(Q`) is a continuous, Zariski dense homomorphism. The continuous weighted completion of Γ relative to ρ is G := ←−lim Gφ φ where φ ranges over all continuous Zariski dense representations Γ → Gφ(Q`) such that (1) Gφ is an affine Q`-group which is a negatively weighted extension of S by a unipotent group:

1 → Uφ → Gφ → S → 1 where H1(Uφ) is negatively weighted;

(2) the composite Γ → Gφ(Q`) → S(Q`) is ρ.

Denote the Lie algebra of the prounipotent radical of the weighted completion G by u. Theorem (Hain-Matsumoto) If Γ is proﬁnite and all S-modules are absolutely irreducible, then ∼ Y 1 ∗ H1(u) = Hcts(Γ, Vλ) ⊗ Vλ ˇ λ∈S,nλ<0 as S-modules, and there is a natural S-equivariant surjection:

Y 2 ∗ Hcts(Γ, Vλ) ⊗ Vλ → H2(u) ˇ λ∈S,nλ<0

Let G` be the Galois group of the maximal algebraic extension of Q that is unramiﬁed away from `. The cyclotomic character × χ` : G` → Z` ,→ Gm(Q`) is continuous and Zariski dense. Theorem (Hain-Matsumoto using Soulé)

The weighted completion A` of G` with respect to χ` is an extension of Gm by the prounipotent group K` with Lie algebra

∧ k` = L(z1, z3, z5, z7,... )

The image of zm in H1(k) has weight −2k. Matsumoto and I used this to prove a conjecture about the 1 action of GQ on the fundamental group of P − {0, 1, ∞} that Ihara attribues to Deligne.

There is a proﬁnite group Γ that is an extension (`) 1 → SL2(Z) → Γ → G` → 1. It is a canonical quotient of the arithmetic mapping class group M of 1,1/Z[1/`] with Tate curve as base point (See Hain-Matsumoto in Lehrer volume of J. Alg. cf. Mochizuki-Tamagawa.) There is a continuous, Zariski dense representation ρ :Γ → GL2(Q`) such that (`) 1 / SL2(Z) / Γ / G` / 1

ρ χ`

det 1 / SL2(Q`) / GL2(Q`) / Gm(Q`) / 1 commutes. Richard Hain Algebraic Completions of Discrete and Profinite Groups Motivation Definition Unipotent Completion Presentation Relative Unipotent Completion Completions of Galois groups The Profinite Case Completion of the arithmetic modular group Weighted Completion Partial computation

Theorem (Hain-Matsumoto) The weighted completion of Γ with respect to ρ is a split extension eis arith 1 → G → G → A` → 1. The Lie algebras of their prounipotent radicals are arith eis eis u = k` n u . The Eisenstein Lie algebra u has minimal presentation

∧ eis M 2n P u = L S (e2n+2) /(r + odd k [zk , r]). n≥1

The ideal r is generated by rf ,m where f is a cusp form of SL2(Z) and m ≥ 2.

Remark 1 Pollack has determined what should be the quadratic relations. They are closely related to classical modular symbols. He has also determined the cubic relations of weight ≤ 30. (Cf. talk at Gusfest last year.) 2 These should imply the relations between depth 2 multiple zeta numbers that come from cusp forms (Gangl-Kaneko-Zagier); they do imply their Galois analogues due to Ihara et al. (Cf. colloquium next week.) 3 Cubic relations should be related to 3-variable modular symbols of classical modular forms. Pollack and I are developing this theory. (Cf. Goncharov, DMJ 2001.)

Richard Hain Algebraic Completions of Discrete and Proﬁnite Groups