Recall
Γ = large arithmetic
� group �
� SL(3, Z), SL 2, Z[α] , etc.
α = irrational, algebraic, real
Some arithmetic groups
that do not act on the circle
University of Lethbridge, Alberta, Canada
http://people.uleth.ca/~dave.morris
Dave.Morris@uleth.ca
Theorem (Carter-Keller-Paige, Lifschitz-Morris)
Proposition (Witte [1994])
Γ does not act on R if Γ � SL(3, Z).
�
�
Lecture 2: Proof for SL 2, Z [α]
using bounded generation
Park City, July 2012
Proof combines bdd generation�and bdd
orbits.
�
�
�
1 ∗
1 0
Unipotent subgroups: U = 0 1 , V = ∗ 1 .
Conjecture
Γ does not act on R. (faithfully — no kernel)
� faithful homomorphism φ : Γ → Homeo+ (R)
Dave Witte Morris
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
Theorem (Lifschitz-Morris [2004])
�
�
Γ does not act on R if Γ � SL 2, Z[α] .
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Theorem (Lifschitz-Morris [2004])
�
�
Γ does not act on R if Γ � SL 2, Z[α] .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
Park City, July 2012
U and V boundedly generate Γ (up to finite index).
Γ acts on R �⇒ U -orbits (and V -orbits) are bdd.
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Bounded generation by unip subgrps
Note: Invertible matrix � Id by row operations.
�
Key fact: g ∈ SL 2, Z) � Id by integer (Z) row ops.
Example
�
�
�
�
�
�
13 31
3 7
3 7
�
�
5 12
5 12
2 5
�
�
�
�
�
�
1 2
1 2
1 0
�
�
�
.
2 5
0 1
0 1
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
Park City, July 2012
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Theorem (Liehl [1984])
�
�
SL 2, Z[1/p] bddly gen’d by elem mats.
I.e., T � Id by Z[1/p] col ops, # steps is bdd.
Proof.
�
�
a b
c d
�
q
�
∗
�
q
�
∗
�
1
�
∗
Park City, July 2012
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
Park City, July 2012
5 / 16
Bdd generation: Γ = U V UV · · · U V .
Bdd orbits: U-orbits and V -orbits are bounded.
q = a + kb prime, p is prim root
�
b
p � ≡ b (mod q); p � = b + k� q
∗
�
p�
p � unit: can add anything to q
∗
�
�
�
�
�
1 0
1 0
p�
�
�
.
∗ 1
0 1
∗
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
Remark: In SL(3, Z), # steps is bounded [Carter-Keller, 1983].
Theorem (Liehl [1984], Carter-Keller-Paige [1995?])
For Z[α] row operations,
#
�
� steps is bounded.
∃n, ∀g ∈ SL 2, Z[α] , g = u1 v1 u�2 v2 · · · u
� n vn .
I.e., U and
V
boundedly
gen
Γ
=
SL
2,
Z[α]
.
�
�
So SL 2, Z[α] = U V U V · · · U V .
∴ U and V generate SL(2, Z).
But # steps is not bounded:
�
U and V do not boundedly generate SL 2, Z).
�
Key fact: g ∈ SL 2, Z) � Id by integer (Z) row ops,
but # steps is not bounded.
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Corollary
φ : Γ → Homeo+ (R) ⇒ every Γ -orbit on R is bdd
⇒ Γ has a fixed point.
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Theorem (Liehl [1984])
�
�
SL 2, Z[1/p] bddly gen’d by elem mats.
I.e., T � Id by Z[1/p] col ops, # steps is bdd.
Easy proof
Assume Artin’s Conjecture:
∀r ≠ ±1, perfect square,
∃ ∞ primes q, s.t. r is primitive root modulo q:
{ r , r 2 , r 3 , . . . } mod q = {1, 2, 3, . . . , q − 1}
Assume ∃q in every arith progression {a + kb}.
∃ q = a + kb, p is a primitive root modulo q.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
Park City, July 2012
6 / 16
Bounded orbits
Theorem (Lifschitz-Morris [2004])
�
�
Γ = SL 2, Z[1/p] acts on R ⇒ every U-orbit bdd.
�
1 u
�
�
�
1 0
�
p
0
�
Assume U-orbit and V -orbit of x not bdd above.
Assume p fixes x.
( p does have fixed pts, so not an issue.)
Proof. Spse ∃ nontrivial action.
It has fixed points:
Remove them:
Take a connected component:
Γ acts on open interval (≈ R) with no fixed pt. →←
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u = 0 1 , v = v 1 , p = 0 1/p
Corollary
Γ cannot act on R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
8 / 16
→←
Wolog u(x) < v(x).
�
�
�
�
Then p n u(x) < p n v(x) .
�
�
LHS = p n u(x) = (p n up −n )(x) → ∞(x) → ∞.
�
�
RHS = p n v(x) = (p n v p −n )(x) → 0(x) < ∞.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
Park City, July 2012
9 / 16
Other arithmetic groups
Open Problem
Show noncocpct arith grps in SL(3, R) and SL(3, C)
cannot act on R.
of higher rank
Proposition
Suppose Γ1 ⊂ Γ2 .
If Γ2 acts on R, then Γ1 acts on R.
If Γ1 does not act on R, then Γ2 does not act on R.
Our methods require Γ to have a unipotent subgrp.
Such arithmetic groups are called noncocompact.
Theorem (Chernousov-Lifschitz-Morris [2008])
Spse Γ is a noncocompact
arith group of higher rank.
�
�
.
Then Γ ⊃ SL 2, Z[α]
or noncocpct arith grp in SL(3, R) or SL(3, C).
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
Park City, July 2012
10 / 16
Optional exercises
6)
�Assume Γ bdd
� gen (by cyclic subgrps).
Show g n | g ∈ Γ has finite index in Γ (∀n ∈ Z+ ).
7) Assume:
Γ1 and Γ2 are arith subgrps of G1 and G2 , resp.
G1 and G2 are simple Lie grps of higher real rank.
Γ1 is cocompact, but Γ2 is not cocompact.
Use the Margulis Superrigidity Theorem to
show Γ2 is not isomorphic to a subgroup of Γ1 .
Park City, July 2012
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V. Chernousov, L. Lifschitz, and D. W. Morris:
Almost-minimal nonuniform lattices of higher
rank, Michigan Mathematical Journal 56, no. 2,
(2008), 453–478.
http://arxiv.org/abs/0705.4330
A. Ondrus: Minimal anisotropic groups of higher
real rank, Michigan Math. J. 60 (2011), no. 2,
355–397.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
Rapinchuk Conjecture implies no action on R
if Γ noncocompact of higher rank.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
Park City, July 2012
Park City, July 2012
16 / 16
(by cyclic subgrps).
If Γ acts by isometries on metric space X,
and every cyclic subgroup has a bdd orbit on X,
then every Γ -orbit on X is bounded.
4)
(harder)
Free group F2 not bdd gen
(if n ≥ 2).
(by cyclic subgrps).
5) U and V do not bddly gen SL(2, Z).
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Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
(Use prev exer.)
Park City, July 2012
12 / 16
Further reading
D. W. Morris: Can lattices in SL(n, R) act on the
circle?, in Geometry, Rigidity, and Group Actions,
University of Chicago Press, Chicago, 2011.
http://arxiv.org/abs/0811.0051
L. Lifschitz and D. Witte: Isotropic
nonarchimedean S-arithmetic groups are not left
orderable, C. R. Math. Acad. Sci. Paris 339 (2004),
no. 6, 417–420.
http://arxiv.org/abs/math/0405536
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
(I.e., Γ = H1 H2 · · · Hn with Hi cyclic.)
3) Γ bdd gen (by cyclic subgrps)
� finite-index subgroup Γ̇ bdd gen.
Cocompact case will require new ideas.
Open Problem
Find cocompact arithmetic group Γ , such that
finite-index subgroups of Γ do not act on R.
1) Assume Γ boundedly generated
2) SL(n, Z) bdd gen by unips
�⇒ SL(n + 1, Z) bdd gen by unips
Related reading
(harder)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
Conjecture (Rapinchuk [∼1990])
These arith grps are boundedly generated by unips.
Exercises
Park City, July 2012
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D. W. Morris: Bounded generation (unpublished).
http://people.uleth.ca/~dave.morris/
banff-rigidity/morris-bddgen.pdf
D. W. Morris: Bounded generation of SL(n, A)
(after D. Carter, G. Keller and E. Paige), New York
J. Math. 13 (2007) 383–421. http:
//nyjm.albany.edu/j/2007/13-17.html
L. Lifschitz and D. W. Morris: Bounded
generation and lattices that cannot act on the
line, Pure and Applied Mathematics Quarterly 4
(2008), no. 1, part 2, 99–126.
http://arxiv.org/abs/math/0604612
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (2/4)
Park City, July 2012
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