What is the
Congruence Subgroup Property?
Dave Witte Morris
University of Lethbridge, Alberta, Canada
http://people.uleth.ca/∼dave.morris
[email protected]
Abstract. We will present a few different characterizations of
the Congruence Subgroup Property, and describe some of its
connections with other important topics in the field of
arithmetic groups, including superrigidity, subgroup growth,
bounded generation, and Algebraic K-Theory.
Dave Witte Morris (Univ. of Lethbridge)
Congruence Subgroup Property
65th Carleton Algebra Day 1 / 15
Theorem (∼1964 Bass-Lazard-Serre, Mennicke)
Every fin-ind subgrp of SLk (Z) is a cong subgrp
if k ≥ 3.
For short, we say that SLk (Z) satisfies the
Congruence Subgroup Property (“CSP”) when k ≥ 3.
65th Carleton Algebra Day 4 / 15
Profinite completion
Example
×
Zp
p
(Zp = the ring of p-adic integers)
Remark (for Γ = SL3 (Z))
( !⇒ Γ # SL3 (Z
()
Z#Z
!⇒ Γ(
$ SL (Z( ).
3
Proposition (assume Γ = SL3 (Z))
()
CSP ⇐⇒ Γ( = SL3 (Z
Dave Witte Morris (Univ. of Lethbridge)
the Z-points of a semisimple algebraic Q-group G
a lattice in a semisimple Lie group G with finite center
Congruence Subgroup Property (CSP):
The finite-index subgroups of Γ are all obvious.
(There are not very many finite-index subgrps.)
Dave Witte Morris (Univ. of Lethbridge)
Congruence Subgroup Property
65th Carleton Algebra Day 2 / 15
So all nonabelian factors in composition series of F
are of the form SL2 (Z/pZ).
However, SL2 (Z) ) free group (up to finite index).
So every finite simple grp is in a composition series.
Dave Witte Morris (Univ. of Lethbridge)
→←
(= × SL3 (Zp ))
p
Congruence Subgroup Property
65th Carleton Algebra Day 7 / 15
!⇒ SL3 (Z/nZ) is finite
Z/nZ is finite
!⇒ H has finite index in SL3 (Z/nZ).
Z → Z/nZ is a ring homomorphism
!⇒ ϕn : SL3 (Z) → SL3 (Z/nZ) is group homo.
−1
Γn,H := ϕn
(H) is a subgroup of finite index in Γ .
Definition
Γn,H is a congruence subgroup of Γ .
These are the obvious finite-index subgroups of Γ .
Dave Witte Morris (Univ. of Lethbridge)
Congruence Subgroup Property
65th Carleton Algebra Day 3 / 15
Congruence Subgroup Property
Proof.
!⇒ Γ( =
65th Carleton Algebra Day 5 / 15
( = × Zp $ Zp .
×
SL3 (Zp ) $ Z
p
p
$ Z.
Note: Kazhdan’s Property T also implies Γ 2
Open question
Can Kazhdan’s property be used to prove CSP?
Congruence Subgroup Property
∀" > 0,
∀ large n
This generalizes CSP to arbitrary abstract groups.
Theorem (1991 Platonov-Rapinchuk)
'
'∗
'
'
Γ
'
'
'
' bdd by a polynomial in n
CSP ⇐⇒
' .g n | g ∈ Γ 0 '
Dave Witte Morris (Univ. of Lethbridge)
Congruence Subgroup Property
65th Carleton Algebra Day 6 / 15
Superrigidity
Lie Theory (p-adic): Lie alg of SL3 (Zp ) is simple,
so no (continuous) homo to abelian grp.
Dave Witte Morris (Univ. of Lethbridge)
Theorem (1995 Lubotzky)
'
"
#
CSP ⇐⇒ # X < Γ '|Γ : X| ≤ n < n" log n
* If infinite, replace with largest of its finite quotients.
Proposition
CSP !⇒ Γ has no infinite abelian quotients.
$Z
The obvious finite-index subgroups of Γ
Let H be any subgroup of SL3 (Z/nZ).
Subgroup growth
Let F = finite quotient of SL2 (Z) = Γ .
k
CSP: F is quotient of SL2 (Z/nZ) ( ×p SL2 (Z/pi i Z)
&
ki
Z/pi Z).
(because Z/nZ (
Γ
CSP: all finite-index subgrps of Γ are obvious
Applications of the CSP
Abelian quotients
CSP calculates the profinite completion of Γ :
Γ
where Γ /N ranges over all
Γ( := lim
the finite quotients of Γ .
←" N
(=
Z
Should also consider other arithmetic groups:
$ √ %
SL2 (Z), SL2 Z[ 5] , Sp4 (Z), . . .
Proof.
Γn,H = congruence subgroup
= obvious finite-index subgroup of Γ
Congruence Subgroup Property
"
#
Let Γ = SL3 (Z) = 3 × 3 integer mats with det 1 .
Theorem
SL2 (Z) does not have the CSP.
Summary
Γ = SL3 (Z) or other arithmetic group
Dave Witte Morris (Univ. of Lethbridge)
Statement of the property
65th Carleton Algebra Day 8 / 15
Theorem (for Γ = SL3 (Z))
ρ : Γ → GLk (R) (any finite-dim’l representation)
!⇒ ρ virtually extends to ρ) : SL3 (R) → GLk (R).
Sketch of proof.
Assume ρ : Γ → SLk (Q).
Γ is finitely generated, so ρ(Γ ) ⊂ SLk (Zp ), ∃p.
extension ρ( : Γ( → SLk (Zp ).
CSP tells us ρ( : SL3 (Zp ) → SLk (Zp ).
Lie theory: any such homomorphism is analytic;
indeed, it is defined by polynomials.
Since ρ(Γ ) ⊂ SLk (Q), the polys have Q coefficients,
so they define homo SL3 (R) → SLk (R).
Dave Witte Morris (Univ. of Lethbridge)
Congruence Subgroup Property
65th Carleton Algebra Day 9 / 15
Proof of the CSP for SL3(Z)
Bounded generation
Recall: Γ finitely generated if ∃ finite set S,
every elt of Γ is product of powers of elts of S.
Definition
Γ is boundedly generated if ∃ finite set S,
every elt of Γ is prod of bdd # of powers of elts of S.
I.e., Γ = .s1 0 · .s2 0 · · · .sn 0.
Conjecture (Rapinchuk)
CSP !⇒ bounded generation.
Proposition (Lubotzky, Rapinchuk)
Converse is true: bounded generation !⇒ CSP.
Dave Witte Morris (Univ. of Lethbridge)
Congruence Subgroup Property
65th Carleton Algebra Day 10 / 15
Key step in proof of CSP: E(nZ) = Γn,{e} .
A starting point of Algebraic K-Theory.
Definition
R = ring
(commutative).
SLk (R)
.
k→∞ Ek (R)
K1 (R) = lim
Exercise
Ek (Z∞ ) = SLk (Z∞ )
(Z∞ = Z × Z × Z × · · · )
!⇒ SLk (Z) is boundedly gen’d (by elem mats).
So methods in pf of CSP apply to prove bdd$gen: %
• short pf of bdd gen of
$ SL3%(Z) and SL
$ 3 Z[α]%
• pf of bdd gen of SL2 Z[α] and SL2 Z[1/r ]
(Actually, easier to use ultrapower instead of ordinary power.)
Dave Witte Morris (Univ. of Lethbridge)
Congruence Subgroup Property
65th Carleton Algebra Day 13 / 15
Proposition (Lubotzky, Rapinchuk)
Bounded generation !⇒ CSP.
Proof.
Γ = .x0 · .y0
Γ
!⇒
= {x, x 2 , . . . , x n } · {y, . . . , y n }.
.nth powers0
'
'
'
'
Γ
'
'
' ≤ n2 is bdd by polynomial.
So '
' .nth powers0 '
Dave Witte Morris (Univ. of Lethbridge)
Congruence Subgroup Property
65th Carleton Algebra Day 11 / 15
CSP for other latts in ss Lie groups
Conjecture (Serre)
Γ = G(Z) arithmetic lattice in conn, ss Lie group G.
(irreducible, G simply conn as alg’ic grp, isotropic at p-adic places)
R-rank G ≥ 2 !⇒ some finite-index subgrp of Γ has CSP.
Remark
True if G/Γ is not compact
[Raghunathan].
Open for some latts in SLk (R) (and others).
Serre conjectured converse:
CSP fails whenever R-rank G = 1.
But lattices in Sp(1, n) or F4,1 may have CSP. (?)
Dave Witte Morris (Univ. of Lethbridge)
Congruence Subgroup Property
65th Carleton Algebra Day 14 / 15
(outline)
• X = finite-index subgroup of Γ = SL3 (Z).
• n = gcd { xij , xii − xjj | x ∈ X }, so
X ⊂ { g ∈ SL3 (Z) | g ≡ λ Id (mod n) } = Γn,{λ Id}

 

* 1 nZ nZ
1
1

 

• E(nZ) =
0 1 nZ , nZ 1
 ⊂ X.
0 0
1
nZ nZ 1
Easy: E(Z) = Γ (= Γ1,{e} ).
Not easy: E(nZ) = Γn,{e} .
(Actually, normal closure of E(nZ) — assume wolog X normal.)
∴ Γn,{e} ⊂ X.
So X ⊃is a congruence subgroup.
Dave Witte Morris (Univ. of Lethbridge)
Congruence Subgroup Property
65th Carleton Algebra Day 12 / 15
A. S. Rapinchuk: The congruence subgroup problem, in:
Algebra, K-theory, groups, and education (New York, 1997).
Amer. Math. Soc., Providence, RI, 1999, pp. 175–188.
MR1732047 (2001f:20108)
B. Sury: The Congruence Subgroup Problem. An elementary
approach aimed at applications. Hindustan Book Agency, New
Delhi, 2003. ISBN 81-85931-38-0, MR1978430 (2005g:20082)
A. Lubotzky and D. Segal: Subgroup Growth. Birkhäuser, Basel,
2003. ISBN 3-7643-6989-2, MR1978431 (2004k:20055)
D. W. Morris: Bounded generation of SL(n, A) (after D. Carter,
G. Keller, and E. Paige). New York J. Math. 13 (2007), 383–42.
MR2357719 (2008j:20145)
Dave Witte Morris (Univ. of Lethbridge)
Congruence Subgroup Property
65th Carleton Algebra Day 15 / 15