An introduction to arithmetic groups (via group schemes) Ste↵en Kionke 02.07.2020 Content Properties of arithmetic groups Arithmetic groups as lattices in Lie groups Last week Let G be a linear algebraic group over Q. Definition: A subgroup Γ G(Q) is arithmetic if it is commensurable to ✓ G0(Z) for some integral form G0 of G. integral form: a group scheme G0 over Z with an isomorphism EQ/Z(G0) ⇠= G. Recall: Here group schemes are affine and of finite type. S-arithmetic groups S: a finite set of prime numbers. 1 ZS := Z p S p | 2 Definition: ⇥ ⇤ A subgroup Γ G(Q) is S-arithmetic if it is commensurable to ✓ G0(ZS) for some integral form G0 of G. - " over bretter : afvm Ks „ t : replace 2 by Ep [ ] Sir= Q by Fpk ) Properties of arithmetic groups Theorem 2: Let Γ G(Q) be an arithmetic group. ✓ * 1 Γ is residually finite. 2 Γ is virtually torsion-free. To Torsion -free % !! with 3 Γ has only finitely many conjugacy classes of finite subgroups. ③ ⇒ finitely many iso classes of finite subgroups Er ⇒ E- finite Es !> leg F isomorph:c to asuboraarp 4 Γ is finitely presented. µ ↳ Fa P is aftype Proof: Γ virtually torsion-free Assume Γ=G0(Z). Claim: G0(Z,b) is torsion-free for b 3. ≥ G0(Z,mby)=ker G0(Z) G0(Z/bZ) ! Suppose g G0(Z,b) has finite order > 1. 2 Vlog ordlg) =p Prime Etbh G : → 2 g- h:&] ? 2-linear Assuwe his out Proof: Γ virtually torsion-free g = " + bh with h: G0 Z onto. O ! ord(g)=p prime. Ik) G. → Go ~> B : OG → & . X h xk ' ' ) Ä = E + E. = = Kribbeln .gl go pbht ⇒ - b 33 : ⇒ tnodb O = b) p p - pbhmodbi ' ' : ⇒ E-omodp o = Eh nodp § M¥3 candida ! Pll! ) fer all #kcp Group schemes and topological groups R : commutative unital ring G : affine group scheme over R . ( . # A : an R-algebra which is also a topological ring. RE, Qp ) Observation: G(A) is a topological group with respect the topology induced by coordinates : G(A) ⇠= V (I ) An c,A ! A c ✓ F. product topdogy : chosen coadiceates . Falt Does not de pend on Group schemes and topological groups A : an R-algebra which is also a topological ring. is onto : → OG Observation: µ Ü OH If ': G H is a closed embedding of affine group schemes over ! R,then ' : G(A) H(A) A ! is a continuous closed embedding of topological groups. thilo G coordinator for H and push ( Hint : pick Ic E Ii GCAIE VIE' ) E VALE ) EHCA ) ) Group schemes and topological groups G: affine group scheme over Z. Consequences: G(R) is a real Lie group (with finitely many connected components). - Closed 9 : E ( G ) Gta enbedding GCR) Gluck) hie ! ¥, group " : real . Varietees have Ehelichen ! Fast alg finite, many " capaats G(Z) G(R) is a discrete subgroup. ✓ " GER) = ) ER K¥2 diente UI UI ↳ " GCZI E Vz (Ic) E Z Theorem of Borel and Harish-Chandra G linear algebraic group over Q Γ G(Q) arithmetic subgroup " ✓ lattice ← " 1 Γ G(R) has finite covolume ✓ () swurjeetine there is no surjective homomorphism G Gm. ( ! GEH ) 2 Γ G(R) is cocompact ✓ () there is no closed embedding Gm G. ! ← G Es anisotropie Remark: Every surjective G Gm splits. ! Examples " RACES Z = Ga(Z) R = Ga(R) is cocompact ✓ ' hom : GET] → Exercise : Thee is no sarjectiue GET ] - of d - algebras GLn(Z) GLn(R) is not a lattice ✓ is det : Glen → Gm SLn(Z) SLn(R) is a lattice but is not cocompact ✓ → S4 a ↳ än swjeetim.no/-cocapact:-Gm[ . " ) laltice : : Sh - Gm - Cf % : SLNHR) → IRX { simple Diagonalization Lemma Let ': Gm GLn be a homomorphism of linear algebraic groups ! over K.Thereisamatrixg GL (K) s.t. 2 n λe1 λe2 1 g'(λ)g− = 0 . 1 .. B C B λen C B C @ A for certain e1,...,en Z and for all λ K⇥. 2 2 Note: If ' is a closed embedding, then e =0for some i. i 6 More examples (1) The Heisenberg group is cocompact: 1 xz H3(Z)= 01y x, y, z Z H3(R) 0 1 | 2 ✓ n 001 o @ A not # 1 are Reasoni.me/ewats , izabk diagonal . More examples (2) F = Q(p2) quadratic number field σ1 : F R with p2 p2 ! 7! σ2 : F R with p2 p2 ! 7! − Observation: (σ1,σ2): F R R E! ⇥ induces an isomorphism R F R R. ⌦Q ! ⇥ More examples (2) Define: T G(A)= g GLn+1(A Z[p2]) g Jg = J { 2 ⌦Z | } p2 − 1 where J = 0 . 1. .. B C B 1C B C @ A Observation: G(R) = O(n, 1) O(n + 1) ⇠ ⇥ {9 ⇐ Glnen (REF ) 153g = )} = { cge.ge/EGLs-ulRIxGLn-nCRI/giJgr=J,gzGB)gz--6dJ) } More examples (2) Claim: G(Z) G(R) ⇠= O(n, 1) O(n + 1) is cocompact. ✓ ⇥ " ← trick a compact factor 9 : Gm --G : - Ä Gm E. (G) Ghana, 9% ,. „ ? " rohes E . ¥ Gült ) izabk with ein diagonal m = 2kt I ) More examples (2) Claim: G(Z) G(R) = O(n, 1) O(n + 1) is cocompact. ✓ ⇠ ⇥ t.iq) - Complex 04+1 ) egnuduaföabs 6<(44-1)--1 ratet . ⇒ Geld Hl) =L Cf is not a closed embedding . An observation Lemma: Let G, H be real Lie groups with finitely many connected components. Let ': G H be a surjective homomorphism with ! compact kernel K =ker('). Assume Γ G is a discrete subgroup, then the following hold: ✓ 1 '(Γ) H is discrete. ✓ 2 Γ torsion-free = Γ = '(Γ). ) ⇠ 3 G/Γ compact H/'(Γ) compact. () 4 Γ G is a lattice '(Γ) H is a lattice. ✓ () ✓ Proof - " Fact: ' is open and proper. ( if Cc ) compact it Cncmpact ) (1) Let h H, U H an open relatively compact neighbourhood. 2 ✓ " " ( Ü ) Z la is compact 4- Q ) Öcalan finite 919in) ni ) = Un 41N rs finite (2) Pn K dis cute ad capaot = , finite " E- ) < es ihn :P 9in Proof Fact: ' is open and proper. (3) “ ”: ' induces is a surjective continuous map ) ': G/Γ H/'(Γ). ! GP h 91g) 9C ) “ ”: If H/'(Γ) is compact, there is a compact set C H with ( ✓ C'(Γ) = H. 1 Then '− (C)Γ= G. and UTC ) maps out % ⇒ compact . Back to the example T Γ=G(Z)= g GLn+1(Z[p2]) g Jg = J 2 | is a discrete cocompact subgroup of G(R) = O(n, 1) O(n + 1). ⇠ ⇥ Project onto first factor: Γ is a discrete cocompact subgroup of O(n, 1). Arithmetically defined groups Definition: Let H be a real Lie group with finitely many connected " " components. ( aithmetie ) → mostly Alattice∆ H is arithmetically defined if ✓ there is a linear algebraic group G over Q, an arithmetic subgroup Γ G(Q) G(R)0 and ✓ \ a surjective homomorphism ': G(R)0 H0 ! with compact kernel such that ∆ and '(Γ) commensurable. Margulis’ arithmeticity Theorem [Margulis]: Let H be a connected simple Lie group such that H = G(R)0 for some linear algebraic R-group G of R-rank 2. ≥ ↳ Closed Every lattice ∆ H is arithmetically defined. ✓ [email protected] A- AutosT Simple groups of R-rank 2 ≥ vk ( Sha ) = n - I SLn(R) for n 3. ≥ rk ( Spzn ) = h Sp (R) for n 2. 2n ≥ ) = / SO(p, q) for p, q 2. vk Cp , q ) tninlpq ≥ SU(p, q) for p, q 2. ≥ Triangle groups 1 1 1 Hyperbolic triangle group: (`, m, n) with ` + m + n < 1. Has a subgroup Γ(`, m, n) of index 2 which is a lattice in + 2 PSL2(R) = Isom (H ) . Takeuchi’s Theorem: Γ(`, m, n) is arithmetically defined if and only if all other roots of the minimal polynomial of λ(`, m, n)=4c2 +4c2 +4c2 +8c c c 4 ` m n ` m n − ⇡ are real and negative (where ck = cos( k )). Arithmetic examples: (2, 3, 7), (2, 8, 8), (6, 6, 6), ... only findet manj Non-arithmetic examples: (2, 5, 7), (3, 7, 7), (4, 11, 13), ... Questions? KEY] Go Chip] ) EGCRIXGCGS ) Questions?.