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. 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 Alternative construction 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. 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 examples 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 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 The unipotent completion of a free group 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) . 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 The unipotent completion of a finitely presented group 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. 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 Minimal presentations of pronilpotent Lie algebras 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. 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 Hodge and Galois structures 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}. 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 Serre’s question 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. 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 Pure braid groups 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. 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 Observations 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.