Introduction to Arithmetic Groups 1 1. Rigidity of linkages Introduction to arithmetic groups What is a superrigid subgroup? 1 rigidity of linkages Example (two joined triangles) Dave Witte Morris 2 group-theoretic superrigidity A A University of Lethbridge, Alberta, Canada 3 the analogy http://people.uleth.ca/ dave.morris ∼ 4 examples of superrigid subgroups [email protected] 5 B B why superrigidity implies arithmeticity D D KAIST Geometric Topology Fair (January 11–13, 2010) C C 6 some geometric consequences of superrigidity Abstract. Arithmetic groups are fundamental groups of locally Mostow Rigidity Theorem A BAC symmetric spaces. We will see how they are constructed, and discuss vanishing of the first Betti number some of their important properties. For example, although the Q-rank of B CBDCD an arithmetic group is usually defined in purely algebraic terms, we will For further reading, see the references in [D. W. Morris, What is a This is not rigid. see that it provides important information about the geometry and superrigid subgroup?, in Timothy Y. Chow and Daniel C. Isaksen, eds.: I.e., it can be deformed (a “hinge"). topology of the corresponding locally symmetric space. Algebraic Communicating Mathematics. American Mathematical Society, technicalities will be pushed to the background as much as possible. Providence, R.I., 2009, pp. 189–206. http://arxiv.org/abs/0712.2299].
Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 1 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 2 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 3 / 64
Example (add a small tetrahedron) A Example (Tetrahedron) A A A BACAD A B CBDCD B D E C B D B D C A tetrahedron is superrigid: the combinatorial B D C E description determines the geometric structure. C A BACAD Combinatorial superrigidity A BACAD B CBDCD B CBDCD B ECEDE Make a copy of the object, This is rigid (cannot be deformed). This is rigid. according to the combinatorial rules. The copy is the exact same shape as the original. However, it is not superrigid: if it is taken apart, it can be reassembled incorrectly. This talk: analogue in group theory.
Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 4 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 5 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 6 / 64
2. Group-theoretic superrigidity Group homomorphism φ: Zk Rd Group homomorphism φ: Z GL(d, R) → → φ extends to a homomorphism φˆ : Rk Rd. (i.e., φ(m n) φ(m) φ(n)) ⇒ → + = · extends to homomorphism φˆ : R GL(d, R). Proof. %⇒ → Group homomorphism φ: Z Rd (Only allow continuous homos.) → Use standard basis of Rk. (i.e., φ(m n) φ(m) φ(n)) e1, . . . , ek ˆ { } + = + ˆ d Define φ(x1, . . . , x ) x φ(e ). Proof by contradiction. φ extends to a homomorphism φ: R R . k = i i ⇒ ˆ → (“linear trans can do anything to a basis") Namely, define φ(x) x φ(1). ! Spse homo φˆ : R GL(d, R) = · Linear transformation ∃ → with φˆ(n) φ(n) for all n Z. Check: homomorphism of additive groups = ∈ ⇒ φˆ(0) Id det φˆ(0) det(Id) 1 > 0 φˆ(n) φ(n) = ⇒ = = = ˆ ˆ ˆ ˆ Group Representation Theory: R connected φ(R" ) connected# φ(x y) φ(x) φ(y) ˆ ⇒ + = + study homomorphisms into Matrix Groups. det φ(R) connected in R× R 0 φˆ is continuous ⇒ ˆ = − { } GL(d, C) d d matrices over C det"φ(R)# > 0 (only allow continuous homomorphisms) = × ⇒ with nonzero determinant. det φ(1) > 0 ⇒ " # This is a group under multiplication. Maybe det φ(1) < 0. " # →← Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 7 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 8 / 64 Dave Witte Morris (Univ. of" Lethbridge)# Introduction to arithmetic groups KAIST Geom Topology Fair 9 / 64 Group homomorphism φ: Z GL(d, R) May have to ignore odd numbers: Lagrange interpolation → extends to homomorphism φˆ : R GL(d, R). restrict attention to even numbers. %⇒ → polynomial curve ∃ n n 1 Because: maybe det φ(1) < 0. y anx an 1x − a0 Analogously, may need to restrict to multiples of 3 = + − + · · · + " # through any n 1 points. 2 (or 4 or 5 or . . . ) + det φ(2) det φ(1 1) det φ(1) > 0. Restrict attention to multiples of . = + = N "In fact,# det φ"(even) ># 0.$ " #% multiples of N is a subgroup of Z { } " # “Restrict attention to a finite-index subgroup" May have to ignore odd numbers: Proposition restrict attention to even numbers. Idea: Zariski closure is like convex hull. Group homomorphism φ: Zk GL(d, R) → φ “almost" extends to homo φˆ : Rk GL(d, R) ⇒ → such that φˆ(Rk) φ(Zk). (“Zariski closure") ⊂ This means Zk is superrigid in Rk.
Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 10 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 11 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 12 / 64
Proposition (“Zk is superrigid in Rk”) 3. The analogy Combinatorial superrigidity Group homomorphism φ: Zk GL(d, R) → Combinatorial superrigidity φ “almost" extends to homo φˆ : Rk GL(d, R) “If the object can be moved somewhere, ⇒ → Make copy of object, obeying combinatorial rules. such that φˆ(Rk) φ(Zk). (“Zariski closure") then the whole universe can be moved there." ⊂ The copy is the exact same shape as the original. is a superrigid subgroup of the group G means: φˆ(Rk) φ(Zk): image of φ controls image of φˆ. ⊂ Maybe not exactly the same object: homomorphism φ: GL(d, R) extends Good properties of φ(Zk) carry over to φˆ(Rk). → may be rotated from the original position; Γ to homomorphism φˆ : G GL(d, R) may be translated from original position. Γ → Example: If all matrices in φ(Zk) commute, These are trivial modifications: Group-theoretic superrigidity then all matrices in φˆ(Rk) commute. rotations and translations are symmetries of the Make a copy of as a group of matrices. 3 Example: If all matrices in φ(Zk) fix a vector v, whole universe (Euclidean space R ). The same copy of can be obtained by moving all then all matrices in φˆ(Rk) fix v. of G into a groupΓ of matrices. Same result can be obtained with the original object Γ Generalize to nonabelian groups. by moving the whole universe to a new position.
Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 13 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 14 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 15 / 64
4. Superrigid subgroups Definition A connected subgroup G of GL(d, C) is solvable if it is upper triangular Example. Zk is superrigid in Rk. Lie groups are of three types: C× CC Generalize to nonabelian groups. solvable (many normal subgrps, e.g., abelian) G 0 C× C ⊂ simple (“no" normal subgroups, e.g., SL(k, R)) 00C× Zk is a (cocompact) lattice in Rk. I.e., combination (e.g., G Rk SL(k, R)) (or is after a change of basis). k = × R is a (simply) connected group (“Lie group") More or less: Zk SL(k, Z) = × Zk is a discrete subgroup ( has a solvable part and a simple part) Example all of Rk is within a bounded distance of Zk Γ All abelian groups are solvable. C, x Rk, m Zk, d(x, m) < C. Today: weΓ consider solvable groups. ∃ ∀ ∈ ∃ ∈ Proof. Every matrix can be triangularized over C. If can replace Zk with and Rk with , G Pairwise commuting matrices then is a (cocompact) lattice in . G can be simultaneously triangularized. Γ Dave Witte Morris (Univ.Γ of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 16 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 17 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 18 / 64 Examples of lattices 1 RC 1 ZZ Zi Proposition + G 0 1 0 0 1 0 superrigid in G G (mod Z(G)). = 0 0 1 = 0 0 1 ⇒ = 1 RRR 1 ZZZ Γ ΓProof. Γ 01RR 01ZZ G G G superrigid G = = The inclusion ! GL(d, R)) = 0 0 1 R = 0 0 1 Z 1 t C 1 RC must extend to G ! GL(d, R) with G . 0 0 0 1 0 0 0 1 Γ ⊂ Γ G, 0 1 0 G 0 1 0 Γ = , = Theorem (converse) G superrigid 00e2πit 00T Γ = Lattice in solv grp G is superrig iff G (mod Z(G)). 1 ZZ Zi is a lattice in = Γ R 00 2Z 00 + + , 0 1 0 . Z = = both G and G,. G : Γsome of the rotations associatedΓ to G, G 0 R+ 0 02 0 0 0 1 %= , = = Z Γ do not come from rotations associated to . 00R+ 0 0 2 Γ so .Γ is not superrigid in . G G, G, G, Γ α 0 Γ = %= %= α α G superrigid E.g., id map : does not extend to ˆ : . rot ∗ | | β Γ = φ φ G, 0 β 0 Γ Γ→ Γ → - . = - β . ( G is abelian but G, is not abelian.) | | = Dave Witte MorrisΓ (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 19 / 64 Dave Witte Morris (Univ. of Lethbridge)Γ IntroductionΓ to arithmetic groups KAIST Geom Topology FairΓ 20 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 21 / 64 Γ Corollary 5. Why superrigidity implies arithmeticity is a superrigid lattice in SL(n, R) A lattice in a Lie group G is “superrigid” iff and every matrix entry is an algebraic number. Let be a superrigid lattice in SL(n, R). G (mod Z(G) (cpct ss normal subgrp)) Γ = · We wish to show SL(n, Z), Γ ⊂ Second, show matrix entries are rational. and simple part of is “superrigid." Γi.e., want every matrix entry to be an integer. Γ Fact. is generated by finitely many matrices. Theorem (Margulis Superrigidity Theorem) Γ Γ First, let us show they are algebraic numbers. Entries of these matrices generate a field extension All lattices in SL(n, R) are “superrigid” if n 3. of Q ofΓ finite degree. “algebraic number field" ≥ Suppose some γi,j is transcendental. Similar for other simple Lie groups, R-rank 2. Then field auto φ of C with φ(γ ) ???. So SL(n, F). For simplicity, assume SL(n, Q) ≥ ∃ i,j = ⊂ ⊂ Corollary (Margulis Arithmeticity Theorem) ab φ(a) φ(b) Define φ . Third,Γ show matrix entries have no denominators.Γ Every lattice in SL(n, R) is “arithmetic" if n 3. -cd. = -φ(c) φ(d). Actually, show denominators are bounded. ≥ ( is like SL(n, Z)) So φ: / GL(n, C) is a group homo. (Then finite-index subgrp has no denoms.) → Only way to make a lattice: take integer points Superrigidity: φ extends to φˆ : SL(n, R) GL(n, C). Since is generated by finitely many matrices, / → (andΓ minor modifications) There are uncountablyΓ many different φ’s, only finitely many primes appear in denoms. / So suffiΓ ces to show each prime occurs to bdd power. Similar for other simple groups with R-rank 2. but SL(n, R) has only finitely many n-dim’l rep’ns. ≥ Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 22 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 23 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 24 / 64
is a superrigid lattice in SL(n, R) Some consequences of superrigidity = superrig fund grp of fin-vol loc symm space . and every matrix entry is a rational number. M ShowΓ each prime occurs to bdd power in denoms. Let be the fundamental group of a locally symmetric space M that has finite volume. ΓCorollary This is the conclusion of p-adic superrigidity: WeΓ always assume M has no compact or flat factors, and is complete. H1(M; R) 0. Also assume M is irreducible: M ≠ M1 M2. = Theorem (Margulis) 0 × Assume is superrigid. Remark If is a lattice in SL(n, R), with n 3, ≥ It is conjectured [Thurston] that if M is a and φ: SL(k, Q ) is a group homomrphism, TopologyΓ of M determines its geometry: → p finite-volume hyperbolic manifold, then thenΓ φ( ) has compact closure. Corollary (Mostow Rigidity Theorem) H1(M; R) ≠ 0, for some finite cover M. I.e., Γ &, no matrix in φ( ) has p& in denom. ∃ If fund group , of finite-vol loc symm space M,, Γ / then is isometric to (up to normalizing constant). So it is believed1 that the fundamental group1 of a Summary of proof: Γ M M, Γ Γ hyperbolic manifold is never superrigid 1 R-superrigidity matrix entries “rational” ⇒ More generally, if ! ,, then M ! M, (although most hyperbolic mflds are Mostow rigid). 2 Q -superrigidity matrix entries Z as a totally geodesic subspace (up to finite covers). p ⇒ ∈ Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 25 / 64 Dave Witte Morris (Univ. of Lethbridge) ΓIntroductionΓ to arithmetic groups KAIST Geom Topology Fair 26 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 27 / 64 Introduction to Arithmetic Groups 2 Examples of arithmetic groups Relation to locally symmetric spaces Recall examples Let be an arithmetic group : X Rn is a symmetric space. relation to locally symmetric spaces M X/ SL(3, Z) = 3 3 integer matrices of det 1 = = Γ = (or subgroup{ of× finite index) } 1 X is homogeneous : Isom(X) is transitive on X compactness criterion (two versions) or SO 1 3; Z SO 1 3 SL 4 Z 1 Γ ( , ) ( , ) ( , ) 1 2 Isom , has an isolated fixed point: Γ 0 = ∩ − 1 φ (X) φ basic group-theoretic properties T I1,3 − 1 ∃ ∈ g SL(4, Z) gI1,3 g I1,3 = − congruence subgroups φ(x) x fixes only 0. Γ ={ ∈ | = } = − residually finite or . . . virtually torsion free (Selberg’s Lemma) Example GZ : G SL(n, Z) for suitable G SL(n, R). finitely presented = = ∩ ⊂ X Hn is also a symmetric space. = (Isom(Hn) SO(1, n)) For further reading, see D. W. Morris, Introduction to Arithmetic Groups. TheoremΓ (“Reduction Theory”) ≈ http://people.uleth.ca/ dave.morris/books/IntroArithGroups.html ∼ For suitable G SL(n, R), is a lattice in G: More advanced: ⊂ Exercise M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer, 1972. 1 is discrete, and X = symmetric space (connected), G Isom(X)◦ V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, 2 Γ = Academic Press, 1993. G has finite volume (maybe compact). # X G/K, K compact. Γ \ ⇒ = Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 28 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 29 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 30 / 64 Γ symmetric space X G/K, K compact locally symmetric space M X, X G/K = = \ = M X, X G/K, a lattice in G = \ = Definition Best mflds are compact. Next best:Γ finite volume. M is a locally symmetric space (complete): Henceforth,Γ assume X has:Γ universal cover of M is a symmetric space. Recall n no flat factors: X ≠ X1 R is a lattice in G: is discrete, and G has finite volume. × I.e., M X, Isom(M) discrete (& torsion-free). \ = \ ⊂ no compact factors: X ≠ X1 compact × Γ Γ Γ Example Γ Γ 0 Proposition Then G is semisimple with no compact factors and trivial center. n n G Isom(R )◦ R ' SO(n), K SO(n) is a torsion-free lattice in G, X G/K = = = = # X Rn. # M X is locally symmetric of finite volume. Fundamental grp provides topological info about any space. ⇒ = ⇒ = \ For locally symmetric spaces, it does much more: Let Zn G, so M Zn Rn Tn. Γ = ⊂ = \ = So latticesΓ are the fundamental groups usually completely determines all of the topology and geometry. ExampleΓ of finite-volume locally symmetric spaces. n Mostow Rigidity Theorem G SO(1, n)◦, K SO(n) # X H . And arithmetic groups are the lattices = = ⇒ = Let SO(1,n; Z), so M Hn = hyperbolic mfld. that are easy to construct. = = \ Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 31 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 32 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 33 / 64 Γ Γ Theorem (Mostow Rigidity Theorem) Compactness criterion 1 Suppose and are finite-volume locally symm. G not compact iff gi, γi, g− γi gi e M1 M2 \ ∃ i → Assume Observation For with : ProofΓ ( ). dim M1 > 2, and M X X G/K ⇐ = \ = Suppose G is compact. is irreducible: (up to finite covers). M is compact iff G is compact. M1 M1 M, M,, \ \ a 1 %≈ × (ΓBecause M G/K and K is compact.) For a, b G, let b a− ba (“conjugation of b by a”). If 1 2, then M1 M2 (modulo a normalizing constant). = \ ∈ = / / Γ Since GΓ is compact, cpct C with C G. \ ∃ = Say is a cocompactΓ lattice in G. Γ EveryΓ aspect of the geometric structure of M C is compact and × is discrete, hence closed, Γ C Γ is reflected as an algebraic property so ( ×) is closed. PropositionΓ of the fundamental group . Γ Therefore G is not compact iff Γ The geometric category of irreducible locally \ 1 g1,g2,g3,... G and e g− γ g ( )G ( ) C symmetric spaces with dimΓ > 2 is equivalent to ∃ ∈ ∈ { i i i} ⊂ × = × Γ C γ1, γ2, γ3,... × e , such that C ΓC . the algebraic category of “irreducible” lattices ∃ 1 ∈ = − { } (( ×) ) ( ×) ( ×) e g− γi gi e. = = Γ = Γ %5 →← in appropriate semisimple Lie groups. i Γ → Γ Γ Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 34 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 35 / 64 Dave Witte Morris (Univ. of Lethbridge)Γ IntroductionΓ to arithmetic groupsΓ KAIST Geom Topology Fair 36 / 64 1 Corollary (Godement Criterion) G not compact iff g , γ , g− γ g e \ ∃ i i i i i → 1 G not compact iff gi, γi, g− γi gi e G not cpct iff has nontrivial unipotent elements. \ ∃ i → \ Corollary Γ Proof ( ). CorollaryΓ (Godement Criterion) Γ Γ SL(2, Z) SL(2, R) is not compact. ⇒ 1 \ g G and γ × with g− γ g Id. G is compact iff has no unipotent elements. ∃ i ∈ i ∈ i i i → \ n Proof. 1 Char poly of Id is det(λ Id) (λ 1) . − = − 2 Γ i 11 ΓDefinition Γ γi SL(n, Z) # char poly of γi has Z coeffs. Let gi and γi . Then n ∈ ⇒ = - 1/i. = - 1. u SL(n, R) is unipotent: (u Id) 0. 3 Similar matrices have same characteristic poly, ∈ − = 1 1 1/i 11 i so the char poly of gi− γi gi also has Z coeffs. g− γi gi i i 1 1/i Equivalently, the only eigenvalue of u is 1, so 4 1 = - .- .- . char poly of g− γ g char poly of Id 1 i i i → 11/i2 10 and both have Z coefficients. e. is conjugate to ... = - 1 . → - 1. = u So the two char polys are equal (if i is large). 0 1 ∗ 5 Therefore, the char poly of is 1 n. γi (λ ) Similarly, SL(n, Z) SL(n, R) is not compact (if n 2). − \ ≥ So the only eigenvalue of γi is 1. Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 37 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 38 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 39 / 64
Corollary (Godement Criterion) Congruence Subgroups Proposition G not cpct iff has nontrivial unipotent elements. is residually finite : \ γ ×, finite-index subgrp H< , γ ∉ H. The proof of the other direction ( ) depends on Γ ∀ ∈ ∃ Γ Γ ⇐ Proof. a fundamental fact from Lie theory: Γ Γ We will use a construction known as γ Id ≠ 0, so (γ Id) ≠ 0. Theorem (Jacobson-Morosov Lemma) − ∃ − ij “congruence subgroups” Choose N % (γ Id)ij γij Idij. If u is any unipotent element of G − = − to prove two basic properties of arithmetic groups. Ring homo Z ZN yields SL(n, Z) SL(n, ZN). (connected semisimple Lie group in SL(n, R)) → → Let ρN : SL(n, ZN) be the restriction to . then there is a continuous homomorphism → 1 is residually finite Since ZN finite, obvious that SL(n, ZN) is finite. 11 ρ : SL(2, R) G with ρ u. 2 is virtually torsion-free Let H ker ρΓ N. Then /H img(ρN) is finite. Γ → - 1. = = / 2 3 Γ So H is a finite-index subgroup. Γ By choice of N, ρN(γ)Γij ≠ ρN(Id)ij, so ρN(γ) ≠ Id. Proof of i So ∉ ker . Let γi u and gi ρ . γ (ρN) H Godement ( ) = ∈ = - 1/i. = ⇐ 2 3 Terminology: ker(ρN ) is a (principal) congruence subgroup of . Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmeticΓ groups KAIST Geom Topology Fair 40 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 41 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 42 / 64 Γ Finite presentation Proposition (Selberg’s Lemma) is finitely generated (if M is compact) is virtually torsion-free : Proposition because S γ γC C ≠ is finite. ={ ∈ | ∩ ∅ } finite-index subgrp H< , H is torsion-free. is finitely presented : Γ ∃ For noncompact case, canΓ construct a nice Γ (no nontrivial elements of finite order) γ1, γ2,..., γm w1,w2, . . . , wr . Γ Γ = 7 | 8 fundamental domain for in X: Proof. ProofΓ of finite generation. C X, γ γC C ≠ is finite. = { ∈ | ∩ ∅ } Define ρ3 : SL(n, Z3) and let H ker(ρ3). Fund grp of any compact mfld is finitely generated: Furthermore, C is open. Γ → = It suffices to show H is torsion-free. π : M M, cpct C M, π(C) M. ΓFinite generationΓ follows from above argument. → ∃ ⊂ = Let h H, writeΓ h Id 3kT , T 0 (mod 3). Let S γ γC C ≠ . ( prop disc, so S finite!) Finite presentation follows from m ∈ k m= + %≡ = 0{ ∈ | ∩ ∅ }0 h (Id 3 T) Then S : a more sophisticated argument = + = 7 8 Γ Id m(3kT) m 32kT 2 Given γ . ΓSince M is connected, chain = + + 2 + · · · ∈ ∃ (since X is connected, locally connected, k $ % k & 1 & Γ Id 3 mT (mod 3 + + ) if 3 m C,γ1C,γ2C,..., γnC γC, with γkC γk 1C ≠ . and simply connected) = ∩ + ∅ ≡ + k & 1 & 1 | Γ 1 1 Id (mod 3 + + ) if 3 + % m. So γ1 S, γ1− γ2 S, ...,γn− γ S. [Platonov-Rapinchuk, Thm. 4.2, p. 195] %≡ ∈ ∈ 1 ∈1 Therefore γ γ1(γ− γ2) (γ− γ) S . = 1 ··· n ∈7 8 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 43 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 44 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 45 / 64 Introduction to Arithmetic Groups 3 Noncompactness via isotropic vectors G SO(Q) and SO(Q; Z). = = n G is not compact v (Q )×, Q(v) 0. Example \ Γ ⇐⇒ ∃ ∈ = noncompactness via isotropic vectors SO(1, 3; Z) is an arithmetic subgroup of SO(1, 3). ProofΓ ( ). = ⇒ Q-rank and the asymptotic cone (provides hyperbolic 3-manifold M H3) Godement Criterion: unipotent u . = \ ∃ ∈ cocompact arithmetic subgroups of SO(1, n) Γ Jacobson-Morosov (over Q): (restriction of scalars) Is M compact? Γ 11 Γ ρ : SL(2, Q) GQ with ρ 1 u. ∃ → 2- .3 = 2 For further reading, see D. W. Morris, Introduction to Arithmetic Groups. Proposition Let a ρ . http://people.uleth.ca/ dave.morris/books/IntroArithGroups.html - 1/2. ∼ & = More advanced: Spse Q(→x) is a (nondegenerate) quadratic form over Z 2 3 V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, 2 2 2 Algebraic Group Thry: a is diagonalizable over Q, (e.g., Q(x1,x2,x3) x1 3x2 7x3) Academic Press, 1993. = − − so has eigenvector Qn with eigenval ≠ 1. Let G SO(Q) and SO(Q; Z). a v λ C. Maclachlan and A. Reid, The Arithmetic of Hyperbolic 3-Manifolds, = = ∈ ± Springer, 2002. Then G is not compact Then Q(v) Q(av) Q(λv) λ2 Q(v). \ ⇐⇒ = = = isotropic Q-vectors:Γ Q(v) 0 with v ≠ 0. Since λ2 ≠ 1, we conclude that Q(v) 0. ∃Γ = = Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 46 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 47 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 48 / 64
G SO(Q) and SO(Q; Z). Example Q-rank and the asymptotic cone = = n & 2 2 2 2 G is not compact v (Q )×, Q(v) 0. Let 7 and SO ; Z . \ ⇐⇒ ∃ ∈ = Q(→x) x0 x1 x2 x3 (Q ) Γ 3= − − − = Then H is compact. (bcs 7 is not a sum of 3 squares) G SO(Q) and SO(Q; Z). ExampleΓ \ = = Γ is not compact Qn , 0. 3 G v ( )× Q(v) SO(1, 3; Z) # H is not compact. Recall:Γ Every positive integer is a sum of 4 squares. \ Γ ⇐⇒ ∃ ∈ = = ⇒ \ So this method will not construct 2 2 2 2 DefinitionΓ ΓProof: Q(x0,x1,x2,xΓ 3) x0 x1 x2 x3, compact hyperbolic n-manifolds for n>3. 1 2= 2− 2− − Let SO(Q; Z). so Q(1, 1, 0, 0) 1 1 0 0 0. = = − − − = Fact from Number Theory V Qn is totally isotropic if Q(V ) 0. Example Γ ⊂ = If Q(x1, . . . , xn) is a quadratic form over Z, Q-rank( ) = max dim tot isotrop Q-subspace. & 2 2 2 2 Let Q(→x) 7x0 x1 x2 x3 and SO(Q; Z). with n 5, and Q is isotropic over R, = 3 − − − = ≥ Then H is compact. then Q is isotropic. Example Γ \ Γ Suppose Q-rank( ) 0. Then ' isotropic Q-vector. Proof: 7 is not a sum of 3 squares (in Q), = Γ& Need to do something more sophisticated. so M X is compact. so has no isotropic vectors. Q(→x) (Return to this later: “restriction of scalars”) = \ Γ Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 49 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 50 / 64 Dave Witte Morris (Univ. of Lethbridge)Γ Introduction to arithmetic groups KAIST Geom Topology Fair 51 / 64
Example Example Theorem (Hattori) Spse SO(1, n), and Hn not compact. Suppose Q-rank( ) 0. Then ' isotropic Q-vector. ⊂ \ Asymptotic cone of X is a simplicial complex = Hn has finitely many cusps. \ so M X is compact. whose dimension is Q-rank( ). = \ Γ\ Γ Look at X fromΓ a large distance. Γ \ Γ Γ More precisely, the asymptotic coneΓ of X is equal to the \ Γ cone on the “Tits building” of parabolic Q-subgroups of G. Γ Remark X compact limit is a point. \ ⇒ X is quasi-isometric to its asymptotic cone. dimension of limit 0 Q-rank( ). Limit = “star" of finitely many rays. \ ∴ = = Γ dimension of limit 1 Q-rank( ). ΓRemark ∴ = = Definition Γ Another important application of Q-rank: asymptotic cone of metric space (M, d) Theorem (Hattori) Γ 1 cohomological dimension of = dim X Q-rank( ) lim (M, d), m0 . Asymptotic cone of X is a simplicial complex − = t t \ (if is torsion-free) →∞ " # whose dimension is Q-rank( ). Γ Γ Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 52 / 64 Dave Witte Morris (Univ. of Lethbridge) IntroductionΓ to arithmetic groups KAIST Geom Topology Fair 53 / 64 Dave Witte MorrisΓ (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 54 / 64 Γ & 2 2 2 2 More general Cocompact lattices in SO(1, n) α √2, Q(→x) x0 αx1 αx2 αx = = − − − ···− n α0, α1,..., αn algebraic integers, s.t. Example Key observation for compactness α0 > 0 and α1, α2,..., αn < 0, Let Galois aut σ of Q(α0,..., α ), Q has no isotropic Q(α)-vectors. ∀ n ασ ,..., ασ all have the same sign α √2, 0 n =& Proof. (all positive or all negative). 2 2 2 2 , Then Q(→x) x0 αx1 αx2 αxn Suppose Q(v) 0. = − − − ···− = SO 2 2 ; R SO 1 , G SO(Q) SO(1, n), Galois auto of Q : σ . G (α0x0 αnxn ) ( , n) = / (α) (a bα) a bα = + · · · + / σ & 2 + 2 =2 − 2 is a cocpct arith subgrp of . GZ G SL n 1, Z[α] . Q (→x) x αx αx αx . GZ[α0,...,αn] G = [α] = ∩ + = 0 + 1 + 2 + · · · + n = 0 Q(v)σ Then is a cocompact" arithmetic subgroup# of G. = Γ (vσ )2 α(vσ )2 α(vσ )2 If n isΓ even, this construction provides all of the = 0 + 1 + · · · + n Later:Γ why is an arithmetic subgroup. Qσ (vσ ). cocompact arithmetic subgroups of SO(1, n). = Since all coefficients of Qσ are positive, Key observationΓ for compactness must have vσ 0. For n odd, also need SO(Q; quaternion algebra) = Q has no isotropic Q(α)-vectors. So v 0. (And n 7 has additional “triality” subgroups.) = = Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 55 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 56 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 57 / 64
Definition of arithmeticity Definition Definition (starting point) G is defined over Q: GZ is an arithmetic subgroup of G Recall G is defined by polynomial funcs with coeffs in Q. if G is defined over Q. Arithmetic subgroup : GZ : G SL(n, Z) for suitable G SL(n, R). Example Complete definition is more general; it ignores: = = ∩ ⊂ SL(n, R) is defined over Q. compact factors of G, and ΓWe always assume G is semisimple, connected. SO(1, n) is defined over Q. differences by only a finite group. & 2 2 2 Theorem Let α √2 and Q(→x) x αx αx . = = 0 − 1 − 2 G is (almost) Zariski closed Then G SO(Q) is not defined over Q. Definition = (defined by polynomial functions on Matn n(R)) (It is defined over Q[α].) Spse G K ! SL(n, R), defined over Q, with K cpct. × × Let , = image of (G K)Z in G. × Example Definition (starting point) Any subgroup of G that is commensurable with , SL 2 R Mat R det 1 GZ is an arithmetic subgroup of G Γ( , has finite index in both and ,) ( , ) A 2 2( ) A ∩ = { ∈ × = } Γ Γ and det A a1,1 a2,2 a41,2 a2,1 is a polynomial. if G is defined over Q. is an arithmetic subgroup of G. = − 4 Γ Γ Γ Γ Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 58 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 59 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 60 / 64
Example & 2 2 2 Restriction of scalars α √2, Q(→x) x αx αx αx2 , = = 0 − 1 − 2 − ···− n SL 2, Z[α] , with α √2, G SO(Q; R), SO Q; Z[α] . = = = = G SL(2, R), = " # " # σΓ = Galois automorphism of Q[α], Want to show Γ is an arithmetic subgroup of G. Recall : G G : γ ( γ, γσ . & 2 2 2 2 → × α √2, Q(→x) x0 αx1 αx2 αx , Then is an arithmetic subgroup of G G. Idea of proof.Γ = = − − − ···− n " # × G SO(Q; R), SO Q; Z[α] . ∆ Γ σ = = Outline∆ of proof. Galois aut of Q(α): (a bα) a bα. Γ σ 2 2+ = 2− Want to show is an" arithmetic# subgroup of G. G SO(x0 αx1 αxn) SO(n 1). Γ Since 1 , α (1, 1), (α, α) is linearly indep, = +σ + · · · + /σ + { }= − 2 Map : z ( (z, z ) gives G G . T ∆ GL∆(2, R) with T Z[α] Z . ⊂ × 6 7 After change of basis (mapping Z n to Z2n), As an warm-up,Γ let us show SL 2, Z[α] is ∃ ∈ =2 2 2 4 ∆ [α] ) / T T T GL(4, R) has T Z[α∆] (Z ) Z . σ " # we∆ have SL(n, Z) Γ (G G )∆ an arithmetic subgroup of SL(2", R) SL#(2, R). = ⊕ ∈ 1 4 4 = 1 = " × So (T T − )(Z ) Z , so T∆ T − SL(4, Z). = ∩ × σ " # so is an∆ arithmetic subgroup of G G . 8 = 8 ⊂ 2 1 ∆ ∆ and TG T − is σ × In fact, 1 SL 4 Z 2 1, Can mod∆ outΓthe compact group G , T8 T8− ( , ) TG8 T − 8 defined over Q Γ = ∩ Γ 2 So is an∆ arithmetic subgroup of G . 8 8 so is arithmeticΓ subgroup of G. 8 8 8 8 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 61 / 64 Dave Witte Morris∆ (Univ.Γ of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 62 / 64 Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 63 / 64 Γ Γ Restriction of scalars Suppose G is defined over Q(α) (algebraic number field), noncon- σ1,..., σ : Q(α) C are the embeddings. n → jugate Then σ1 σ2 σ G∗ G G G n is defined over Q, = × × ···× and
GZ[α] is isomorphic to (G∗)Z via the map : γ ( (γσ1 , γσ2 ,..., γσn ).
We assume here that Z[α] is the entire ring of integers of Q(α). ∆ & Example: If Q(→x) has coefficients in Q(α), then SO Q; Z[α] is an arithmetic subgroup of a" product# of orthogonal groups.
Dave Witte Morris (Univ. of Lethbridge) Introduction to arithmetic groups KAIST Geom Topology Fair 64 / 64