An introduction to arithmetic groups (via group schemes)

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

LetGbe a linear algebraic group over Q. Definition:

A subgroup ✓G(Q) is arithmeticif it is commensurable to G0(Z) for some integral formG0 of G.

integral form: a group scheme G0 overZ with an isomorphism E_Q/Z(G₀)⇠=G.

Recall: Here group schemes are affine and of finite type.

S-arithmetic groups

S: a finite set of prime numbers.

ZS :=Z⇥1

p |p2S⇤ Definition:

A subgroup ✓G(Q) is S-arithmeticif it is commensurable to G₀(ZS) for some integral formG₀ of G.

-

bretter^:

„

afvm

^Ks

"

Sir

^replace

^Ep

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.

3 has only finitely many conjugacy classes of finite subgroups.

4 is finitely presented.

*

%

! ^!

^free

③

finitely

of finite

E-^Er

finite

to asuboraarp

µ

↳

aftype ^Fa

Proof: virtually torsion-free

Assume =G₀(Z).

Claim: G₀(Z, b) is torsion-free forb 3.

G0(Z, m) = ker G0(Z)!G0(Z/bZ) Supposeg2G0(Z, b)has finite order >1.

by

Vlog ordlg

G

^{2 g- Etbh}

h

:&

2-linear Assuwe his out

Proof: virtually torsion-free

g="+bh with h:OG0 !Zonto.

ord(g) =p prime.

Ik)

G. ^→

Go

OG

&

X ^h xk

E. =

.gl

^{go Ä}

'

= E +

pbht Kribbeln

tnodb _b)

-

: O =

pbhmodbi

M¥3

^{Eh nodp}

^E-

^§

Pll ^{! ) fer}

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.

Observation:

G(A) is a topological groupwith respect the topology induced by coordinates

c,A:G(A) ^⇠=!V_A(I_c)✓Aⁿ

(

Qp

⁾

F.

product topdogy

Falt

pend

Group schemes and topological groups

A:an R-algebra which is also a topological ring.

Observation:

If':G!H is a closed embeddingof affine group schemes over R, then

'_A:G(A)!H(A)

is a continuousclosed embedding of topological groups.

µ

Ü

OH

(^{Hint :} pick

coordinator for

thilo

Ic

Ii

GCAIE

VIE

^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).

G(Z)✓G(R)is a discrete subgroup.

9

^enbedding

GCR)

¥

group !

Fast^{: real} alg^{. Varietees have} finite,many ^"Ehelichen^"capaats^!

GER) =

K¥2

UI UI ^↳diente

GCZI ^E

Vz

Theorem of Borel and Harish-Chandra

Glinear algebraic group over Q

✓G(Q)arithmetic subgroup

1 ✓G(R) has finite covolume ()

there is nosurjective homomorphismG!Gm.

2 ✓G(R) is cocompact ()

there is noclosed embeddingGm!G.

Remark: Every surjective G!G^m splits.

←^"

lattice

(

GEH )

← G Es

anisotropie

Examples

Z=Ga(Z)✓R=Ga(R) is cocompact

GL_n(Z)✓GL_n(R) is not a lattice

SL_n(Z)✓SL_n(R) is a lattice but is not cocompact RACES^"

Exercise^: Thee is ^no sarjectiue ^{hom :} GET^{] →} GET^']

- of^d-algebra^s

det

Gm

swjeetim.no/-cocapact:-Gm

_[ ^än

."

)

laltice ^:

Cf

-

%

^IRX

{ simple

Diagonalization Lemma

Let':Gm !GL_n be a homomorphism of linear algebraic groups overK. There is a matrixg2GL_n(K) s.t.

g'( )g ¹ = 0 BB B@

e1

e2

. ..

en

1 CC CA

for certaine₁, . . . , e_n2Z and for all 2K^⇥.

Note: If'is a closed embedding, then e_i 6= 0 for somei.

More examples (1)

The Heisenberg group is cocompact:

H₃(Z) =n0

@1 x z 0 1 y 0 0 1

1

A|x, y, z2Zo

✓H₃(R)

Reasoni.me/ewats

diagonal

More examples (2)

F =Q(p

2)quadratic number field

1:F !R with p

27!p 2

2:F !R with p

27! p 2

Observation:

( 1, 2) : F !R⇥R induces an isomorphismR⌦_QF^E!R⇥R.

More examples (2)

Define:

G(A) ={g2GL_n+1(A⌦ZZ[p

2])|g^TJg =J}

whereJ = 0 BB B@

p2 1

. ..

1 1 CC CA.

Observation: G(R)⇠= O(n,1)⇥O(n+ 1)

{

⁽ _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.

← a

compact factor

9

^Gm

G

9%

^Gm

^{(G) Ä Ghana}

Gült

diagonal

^E

^¥

"

m= 2ktI)

More examples (2)

Claim: G(Z)✓G(R)⇠= O(n,1)⇥O(n+ 1) is cocompact.

t.iq ₎

-

04+1

) Complex

egnuduaföabs

6<(44-1)--1

Geld

) ^=L

Cf

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 kernelK = 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.

(1) Leth2H,U ✓H an open relatively compact neighbourhood.

(2)

( if

^{Cncmpact )}

compact

4-

^Q

Öcalan

^finite

919in

⁾

^41N

^finite

Pn K

capaot

finite

<

"

es

ihn ^{:P E- 9in}

Proof

Fact: 'is open and proper.

(3) “)”: ' induces is a surjective continuous map ':G/ !H/'( ).

“(”: IfH/'( )is compact, there is a compact set C ✓H with C'( ) =H.

Then' ¹(C) =G.

GP

91g

and

UTC ⁾

%

compact

Back to the example

=G(Z) = g2GL_n+1(Z[p

2])|g^TJg=J

is a discrete cocompact subgroup ofG(R)⇠= O(n,1)⇥O(n+ 1).

Project onto first factor:

is a discrete cocompact subgroup of O(n,1).

Arithmetically defined groups

Definition:

LetH be a real Lie group with finitely many connected components.

A lattice ✓H is arithmetically definedif there is a linear algebraic group GoverQ, an arithmetic subgroup ✓G(Q)\G(R)⁰ and a surjective homomorphism ':G(R)⁰ !H⁰ with compact kernel

such that and'( )commensurable.

→ (mostly ^"aithmetie"

)

Margulis’ arithmeticity

Theorem[Margulis]:

LetH be a connected simple Lie group such that

H=G(R)⁰ for some linear algebraic R-group Gof R-rank 2.

Every lattice ✓H is arithmetically defined.

↳ Closed

embeddagGic-sG@verRIFaeg.Htririalcah ^Ah

Autos ^T

Simple groups of R -rank 2

SL_n(R) for n 3.

Sp_2n(R) for n 2.

SO(p, q) forp, q 2.

SU(p, q) for p, q 2.

vk (Sha ) ^{= n- I} rk

( Spzn

vk Cp^{, q)}

)

Triangle groups

Hyperbolic triangle group: (`, m, n) with <sup>1</sup><sub>`</sub> +_m¹ +_n¹ <1.

Has a subgroup (`, m, n)of index 2 which is a lattice in PSL2(R) = Isom⁺(H²) .

Takeuchi’s Theorem:

(`, m, n)is arithmetically defined if and only if all other roots of the minimal polynomial of

(`, m, n) = 4c<sup>2</sup><sub>`</sub>+ 4c²_m+ 4c²_n+ 8c_`c_mc_n 4

are real and negative (wherec_k= cos(^⇡_k)).

Arithmetic examples: (2,3,7),(2,8,8),(6,6,6),. . . Non-arithmetic examples: (2,5,7),(3,7,7),(4,11,13),. . .

only findet

j

Questions?

KEY ^{] Go Chip}

^{) EGCRIXGCGS )}

Questions?