Recall is a lattice in G means Why arithmetic groups are lattices is a discrete subgroup of G, and Γ G/ has finite volume (e.g., compact) Dave Witte Morris Γ RemarkΓ University of Lethbridge, Alberta, Canada http://people.uleth.ca/ dave.morris M = hyperbolic 3-manifold of finite volume (e.g., compact) ∼ M h3/ , where = (torsion-free) lattice in SO(1, 3). [email protected] ⇐⇒ = More generally: Γ Γ Abstract. SL(n, Z) is the basic example of a lattice in SL(n, R), and a locally symmetric space of finite volume lattice in any other semisimple Lie group G can be obtained by (torsion-free) lattice in semisimple Lie group G intersecting (a copy of) G with SL(n, Z). We will discuss the main ideas ←→ behind three different approaches to proving the important fact that the integer points do form a lattice in G. So it would be nice to have an easy way to make lattices. Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 1 / 19 Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 2 / 19 Example Theorem (Siegel, Borel & Harish-Chandra, 1962) SL(n, Z) is a lattice in SL(n, R) G is a (connected) semisimple Lie group G ! SL(n, R) SL(n, Z) is the basic example of an arithmetic group. G is defined over Q i.e. G is defined by polynomial equations with Q coefficients More generally: i.e. Lie algebra of G is defined by linear eqs with Q coeffs For G SL(n, R) (with some technical conditions): i.e. GQ is dense in G ⊂ i.e. is Zariski dense in [if has no compact factors] GZ G SL(n, Z) is an arithmetic subgroup of G. GZ G G = ∩ i.e. GZ " connected, proper subgroup of G GZ is a lattice in G. # GZ is a lattice in G. ⇒ Arithmetic groups are lattices. Remark Remark (“Margulis Arithmeticity Theorem”) It is obvious that GZ is discrete. Converse if R-rank G 2 (and is irreducible). It is not obvious that G/GZ has finite volume. ≥ Γ Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 3 / 19 Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 4 / 19 GZ is a lattice in G A short proof for some lattices Example Example SL 2 Z is a lattice in SL 2 R . ( , ) ( , ) SO(n, 1; Z) is a lattice in SO(1, n). Not so easy to prove. Proof. Example Fundamental domain: 2 2 2 2 B(x1, . , x4) x x x 7x = 1 + 2 + 3 − 4 # SO B(x)" ; Z) is a lattice in SO B(x)" SO(3, 1). ⇒ + In fact,# G/GZ is compact. # $ Proposition G SO B(x)" = B(x)" a# (nondegenerate)$ quadratic form with Z coeffs n B(x)" anisotropic: B(v) ≠ 0 for v (Z )× 1/2 ∞ dx dy ∞ 1 ∈ Area ( )< 2 dy < # G/GZ is compact. F ! 1/2 y y = ! y ∞ " "− " ⇒ Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 5 / 19 Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 6 / 19 Proof For Part 2 of the proof, we need a lemma: G/GZ ! SL(n, R)/ SL(n, Z) Mahler Compactness Criterion Proof has 2 parts: Let C SL(n, R). ⊂ 1 The image of C in SL(n, R)/ SL(n, Z) is precompact G/GZ is closed in SL(n, R)/ SL(n, Z). 0 is not an accumulation point of CZn. (True for any G defined over Q.) ⇐⇒ 2 G/GZ is precompact in SL(n, R)/ SL(n, Z). Proof ( ). ⇒ Suppose c z 0 and c γ h. We may assume γ e. n n → n n → n = Part 1 of the proof. Then hz c z 0. But hZn is discrete. So z 0. n ≈ n n ≈ n = Suppose g γ h with g G and γ SL(n, Z). i i → i ∈ i ∈ Converse is an exercise. (Try for n 2 first.) For v Zn: B(γ v) B(g γ v) B(hv), = ∈ i = i i → but B(γ v) B(Zn) Z. i ∈ ⊂ Part 2 of the proof. So is (eventually) constant: 0 . B(γiv) B(hv) B(γiv) B(γ v) n n = = gnvn G(Z )× # B(gnvn) B(vn) B (Z )× Z× For simplicity, assume γ0 e. ∈ ⇒ = ∈ ∈ = (because B(x)" is anisotropic with Z coefficients) Then B(hv) B(γiv) B(γ0v) B(v). # $ = = = # B(g v ) 0 So h SO B(x)" G. ⇒ n n /→ ∈ = # gnvn 0. # $ ⇒ /→ Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 7 / 19 Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 8 / 19 A dynamical proof that GZ is a lattice [Margulis] Corollary u-invariant ϕ L1 SL(n, R)/ SL(n, Z) % 0 . ∃ ∈ { } Any line segment that reaches the # $ boundary of the large disk has only a Proof. small fraction of its length inside the SL(n, R)/ SL(n, Z) can be covered by countably many cpct sets, tiny disk. so cpct C that works for all x in a set of positive measure. ∃ # k [0,R] ukx C Let ϕ(x) lim infR { ∈ | ∈ } . = →∞ R 1 1 2 R Theorem (Dani-Margulis) ϕ lim inf χ χ u− χ u− χ u− G/ = R C + C ◦ + C ◦ + · · · + C ◦ 1 1 2 R SL R SL Z , % lim% inf #χC χC u− χC u− χC u− $ x (n, )/ (n, ) Γ ≤ G/ R + ◦ + ◦ + · · · + ◦ ∈ (“Fatou’s Lemma”) u SL(n, R) unipotent % Γ # $ ∈ lim inf 1 1 R I.e., is conjugate to an element of . R χC χC u− χC u− u N = 1 + ◦ + ◦ lim inf #% χC % χC χC % χC$ # compact SL R SL Z , = R + + + · · · + C (n, )/ (n, ) χ ⇒∃ ⊂ = C #% % % % $ < . # k [0, R] ukx C ∞% lim inf { ∈ | ∈ } > 0 And 0 on a set of positive measure, so ≠ 0 in 1. R ϕ > ϕ L →∞ R Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 9 / 19 Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 10 / 19 Combine with decay of matrix coefficients: Corollary Theorem (Howe-Moore) SL(n, Z) is a lattice in SL(n, R). ϕ, ψ L2(G/ ) constants # lim ϕ, ψ g 0. g ∈ 3 ⇒ →∞4 ◦ 5 = Proof. Γ We know u-invariant ϕ L1 SL(n, R)/ SL(n, Z) % 0 . In fact, valid for vectors in any unitary representation of G. ∃ ∈ { } Howe-Moore: ϕ is constant. # $ Corollary (Mautner Phenomenon) So SL(n, R)/ SL(n, Z) has finite measure. H noncompact, closed subgroup of G, ϕ Lp(G/ ), H-invariant Proof generalizes easily to general GZ. ∈ Just need to know G/GZ is closed in SL(n, R)/ SL(n, Z). # ϕ is constant. ⇒ Γ Chevalley’s Theorem: G defined over Q Proof. # ϕ : SL(n, R) SL(N, R) with ⇒∃ → Wolog assume p 2 and ϕ constants. G Stab(vector in QN ), and = ⊥ 2 = . 0 lim ϕ, ϕ g lim ϕ, ϕ h lim ϕ, ϕ ϕ 2 = g 4 ◦ 5 = h 4 ◦ 5 = h 4 5 = 7 7 ϕ SL(n, Z) SL(N, Z). →∞ →∞ →∞ ⊂ # ϕ 0 is constant. ⇒ = # $ Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 11 / 19 Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 12 / 19 Proof of Howe-Moore Theorem. The traditional proof Let g G, with g . { j} ⊂ 7 j7→∞ For simplicity, assume G SL(2, R) and gj ∗ A. = { } ⊂ & ∗' = Passing to subseq, wma gj converges weakly, to some E: 2 Example gj ϕ ψ Eϕ ψ ϕ, ψ L . 4 | 5→4 1 | 5∀ ∈ 1 SL(n, Z) is a lattice in SL(n, R). Also assume gjug− e for u N ∗ . So j → ∈ = & 1' 1 Euϕ ψ lim g uϕ ψ lim (g ug− )g ϕ ψ 4 | 5 = 4 j | 5 = 4 j j j | 5 Proof uses Iwasawa decomposition: G KAN, where lim g ϕ ψ Eϕ ψ , = = 4 j | 5 =4 | 5 K SO(n), so Eu E; i.e., E(u Id) 0. = = − = % u-inv’t vectors in L2 # img(u Id) dense, so E 0. A = diagonal matrices (with positive entries, so A is ⇒ − = connected), Actual proof: N = upper triangular, with 1s on the diagonal. E annihilates img(u Id). (To show g k a n, use Gram-Schmidt orthogonalization.) − 1 = E∗ annihilates img(v Id) for v N− 1 . − ∈ = &∗ ' E∗E EE∗, so ker E ker E∗. = = img(u Id) img(v Id) dense in L2 constants. − + − 3 Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 13 / 19 Dave Witte Morris (Univ. of Lethbridge) Why arithmetic groups are lattices U of Chicago (June 2010) 14 / 19 a11(g) Example a22(g) Write g k a n, with a a(g) . For n 2, let .. = = = = ann(g) t Ac A 0 <t c Note that a11(g) ge1 . = (& 1/t' ∈ ) ≤ * = 7 7 ) ) 1 t ) Key Lemma Nc N )t c = (&01' ∈ ) | | ≤ * a11(g) a11(gγ), γ SL(n, Z) # a11(g) < 2 a22(g). ) ≤ ∀ ∈ ⇒ (“Siegel set”) S KA) N c = )c c 2 ) Corollary The picture of Sc in h is: For n 2: S SL(2, R) if c √2.