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 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 ane and of finite type. S-arithmetic groups

S: a finite set of prime numbers.

1 ZS := Z p S p | 2 Definition: ⇥ ⇤ A subgroup 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 G(Q) be an arithmetic group. ✓ * 1 is residually finite.

2 is virtually torsion-free. To Torsion -free % !! with 3 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 is finitely presented. µ ↳ Fa P is aftype Proof: virtually torsion-free

Assume =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: 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 : ane 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 ane 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: ane 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

G(Q) arithmetic subgroup " ✓ lattice ← " 1 G(R) has finite covolume ✓ () swurjeetine there is no surjective homomorphism G Gm. ( ! GEH )

2 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 e1 e2 1 g'()g = 0 . 1 .. B C B en C B C @ A for certain e1,...,en Z and for all 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

1 : F R with p2 p2 ! 7! 2 : F R with p2 p2 ! 7!

Observation: (1,2): 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 G is a discrete subgroup, then the following hold: ✓

1 '() H is discrete. ✓

2 torsion-free = = '(). ) ⇠

3 G/ compact H/'() compact. ()

4 G is a lattice '() 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/ H/'(). ! GP h 91g) 9C ) “ ”: If H/'() is compact, there is a compact set C H with ( ✓ C'() = H.

1 Then ' (C)= G.

and UTC ) maps out %

⇒ compact . Back to the example

T =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: 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 G(Q) G(R)0 and ✓ \ a surjective homomorphism ': G(R)0 H0 ! with compact kernel such that and '() 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 (`, m, n) of index 2 which is a lattice in + 2 PSL2(R) = Isom (H ) . Takeuchi’s Theorem: (`, m, n) is arithmetically defined if and only if all other roots of the minimal polynomial of

(`, 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?