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Lattices and Nilmanifolds Chapter 2 Lattices and nilmanifolds In this chapter, we discuss criteria for a discrete subgroup Γ in a simply connected nilpotent Lie group G to be a lattice, as well as properties of the resulting quotient G=Γ. 2.1 Haar measure of a nilpotent Lie group The map exp is a diffeomorphism between the Lie algebra g of a simply connected Lie algebra G and G itself. On both g and G there are natural volume forms. On g, this is just the Euclidean volume denoted by mg. On G, there is a natural left invariant measure, the left Haar measure, denoted by mG. The choices of mg and mG are unique up to a renormalizing factor. Proposition 2.1.1. Suppose G is a simply connected nilpotent Lie group and g is its Lie algebra. After renormalizing if necessary, the pushforward measure exp∗ mg coincides with mG. r Proof. Fix a filtration fgigi=1 (for example the central lower series fg(i)g) and a Mal'cev basis X adapted to it. Since the left Haar measure is unique to renormalization, it suffices to show that exp∗ mg is invariant under left multiplication. This is equivalent to that mg is invariant under left multi- plications in the group structure (g; ). On g, use the linear coordinates (1.16) determined by X . Then for Pm Pm U = j=1 ujXj and V = j=1 VjXj, W = U V is given by the formula (1.18). Thus the partial derivative in V of U V can be written in block 32 CHAPTER 2. LATTICES AND NILMANIFOLDS 33 form as 0 Id ∗ · · · ∗ 1 B Id ∗ C B C ; B .. C @ . A Id where the i-th block correspond to the vi component. Since the determinant of this matrix is 1, the map V ! U V preserves the Euclidean volume mg. This completes the proof. Observe that the partial derviative in U of U V has the same block form as above, hence mg is also right invariant in (g; ). So we get: Corollary 2.1.2. Every nilpotent Lie group G is unimodular, i.e. its left and right Haar measures coincide up to rescaling. Proof. If G is simply connected, the remark above asserts that mG = exp∗ mg is both left invariant and right invariant. This yields the unimod- ularity as desired. For a connected nilpotent Lie group G, its universal cover Ge is a Lie group with the same Lie algebra g and hence, by Theorem 1.5.7, is nilpotent as well. The left and right Haar measures of Ge coincide and descend to the corresponding Haar measures on G, which still coincide (up to rescaling). For a disconnected nilpotent Lie group G, let G0 be its identity compo- nent, which is still nilpotent and is thus unimodular. It suffices to note in the case the product measure between the counting measure on G=G0 and mG0 is both left and right invariant on G. Exercises Exercise 2.1.1. Describe the Haar measure of the group of upper triangular unipotent d × d matrices. Exercise 2.1.2. Show that for d ≥ 2, the group of upper triangular d × d matrices with positive diagonal entries is not unimodular. 2.2 Lattices Definition 2.2.1. Given a connected Lie group G, a discrete subgroup Γ ⊆ G is called a lattice if the quotient G=Γ admits a finite measure that is left invariant by elements of G. CHAPTER 2. LATTICES AND NILMANIFOLDS 34 Lemma 2.2.2. If G has a lattice, then G is unimodular. Proof. Because Γ is discrete, one can lift the left invariant meaure mG=Γ to a left invariant measure mG by making the projection G ! G=Γ a locally volume preserving map. For every h 2 G,(Rh)∗mG is also left invariant, where Rh is the right multiplication by h. Because left Haar measure is unique up to rescaling, (Rh)∗mG = χ(h)mG for a function χ : G ! R>0. Note that χ is a group morphism into the multiplicative group R>0. Since the lifting is equivariant under the deck transformation group Γ, Γ ⊆ ker χ. The pushforward µ of mG=Γ from G=Γ to R>0 by g ! χ(g) is invariant under multiplication by χ(k) for every k 2 G. Unless χ ≡ 1, which means mG is also right invariant, the measure µ on R>0 in invariant under multi- plicationby λ for some λ > 1. Such a measure cannot have finite totall mass, though µ(R>0) = mG=Γ(G=Γ) < 1. This yields a contradiction. Hence mG is right invariant and G is unimodular. n n Example 2.2.3. Z is a lattice of R . Hereafter, G will be assumed to be a simply connected nilpotent Lie group. We will identify G with its Lie algebra g via the exponential map exp. Using this identification, the group structure can be thought of as (g; ) where is from (1.11). We remark that: Proposition 2.2.4. Every simply connected nilpotent Lie group G is an algebraic group. That is, using the linear coordinates (1.16) of g, G is identified with the 1 ∼ m affine algebraic variety g = R and the group operations, which are the multiplication and the inversion X ! −X, are polynomial. Since every connected closed subgroup H is identified with its Lie algebra h, which is a subspace of g. It is clear that: Lemma 2.2.5. In Proposition 2.2.4, every connected closed subgroup is a Zariski closed algebraic subgroup. The main result of this section is: Theorem 2.2.6. Let G be a simply connected nilpotent Lie group endowed with the algeraic structure of its Lie algebra g, and Γ ⊂ G be a discrete subgroup, then the following are equivalent: 1Algebraic subvarieties of g are the zero sets of finite arrays of polynomial equations. They are the closed subsets of a topology, the Zariski topology. CHAPTER 2. LATTICES AND NILMANIFOLDS 35 (1) Γ is a lattice; (2) Γ is Zariski dense; (3) Γ is not contained in any proper connected closed subgroup of G; (4) exp−1 Γ is not contained in any proper vector subspace of g. (5) Γ is cocompact, i.e. G=Γ is compact. Proof. We shall show (5))(1))(2))(4))(3))(2))(5). (5))(1). Because G is unimodular by Corollary 2.1.2, there is a bi-invariant Haar measure mG, which projects to a measure mG=Γ on G=Γ by right invariance. Moreover mG=Γ inherits the left invariance property. The compactness of G=Γ means that there is a compact set Ω ⊂ G such S that γ2Γ Ωγ = G. Then mG=Γ(G=Γ) ≤ mG(Ω) < 1. So Γ is a lattice. (1))(2). Let H be the Zariski closure of Γ, i.e. the smallest closed algerbaic subvariety of G containing Γ. Then as a standard fact about topological groups, H is a subgroup of G. (Exercise 2.2.1) Let H0 be the identity component of H in Zariski topology. Then H0 is a subgroup of H and has finite index. Since every Zariski connected closed subvariety is connected and closed in the usual Hausdorff topology. H0 is a connected closed subgroup of G. By Lemma 1.5.16, H = H0. In other words, H is Zariski connected and connected. By Theorem 1.1.18, H is a Lie subgroup, whose Lie algebra is h := exp−1 H. Assume for contradiction that H ( G, then h ( g. Let g(2) = [g; g] be the commutator subalgebra of g and G(2) = exp g(2). Then g(2) is an ideal and G(2) is a connected closed normal subgroup of G. Because of normality, HG(2) is a subgroup of G, which is again connected and closed and hence a Lie subgroup. Its Lie subalgebra is h + g(2). By Lemma 1.4.12, h + g(2) 6= g as h 6= g. So HG(2) = exp(h + g(2)) is a proper subgroup of G. Moreover, h + g(2) is an ideal of g, because [g; h + g(2)] ⊆ [g; g] = g(2) ⊆ h + g(2): Thus HG(2) is a normal subgroup in G by Lemma 1.5.4. Consider the quotient group G=HG(2), which is non-trivial by the para- graph below, abelian as it is a quotient of G=G(2), which is abelian, and simply connected by Corollary 1.5.14. Thus there exists d ≥ 1 such that d G=HG(2) is isomorphic to R as a group. ∼ d Take the pushforward measure µ = τ∗mG=Γ on G=(HG(2)) = R by the map τ : g ! gHG(2). (Note that τ is well-defined on G=Γ since Γ ⊂ HG(2).) Then µ has finite volume and is invariant under left translations, which is d impossible on R . The desired contradiction is hence obtained. CHAPTER 2. LATTICES AND NILMANIFOLDS 36 (3))(2). Again, let H be the Zariski closure of Γ. Then as above, H must be a connected closed subgroup of G. So by assumption, H = G, which means that Γ is Zariski dense in G. (2))(4). This implication is obvious because the Zariski topology of G is inherited from that of g via exp, and a proper vector subspace is a proper algebraic subvariety. (4))(3). If Γ ⊆ H for a proper connected closed subgroup H of G, then exp−1 Γ ⊆ h, which is a proper vector subspace of g.
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