Some arithmetic groups
that do not act on the circle
Question
¿ ∃ (faithful) action of Γ on X ?
University of Lethbridge, Alberta, Canada
http://people.uleth.ca/~dave.morris
Dave.Morris@uleth.ca
Question
¿ ∃ (faithful) action of Γ on R ?
Example
SL(2, Z) does not act on R.
Proof.
�
1 / 16
−1 0 2
= I. So SL(2, Z) has elt’s of finite order.
0 −1
Example
Γ � SL(2, Z) finite-index subgrp can be a free group.
Has many actions on R.
Park City, July 2012
4 / 16
� left-inv’t order on Γ � SL(3, Z).

1 1 0

1 0,
1

y=


1 0 0

1 1,
1
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
Conjecture
Large arithmetic groups
or Homeo+ (S )?
Park City, July 2012

z=

[“lexicographic order"])
7 / 16
�
�
irreducible,
R-rank > 1
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
Wolog x, y, z � e.

1 0 1

1 0
1
Park City, July 2012
(because Γ̇

1

0
0
Z
1
0
2 / 16
� Z)
(⊂ SO(1, n)?)
cannot act on R
Park City, July 2012
Γ acts on S 1 − (fixed pt)
≈ R.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
Park City, July 2012
Algebraic translation of conjecture
Definition
Assume Γ acts (faithfully) on R.
a≺b
⇐⇒
a(0) < b(0) or . . .
Note: a, b � e �⇒ ab � a � e and e � a−1 .
Exercise (assume Γ countable)
Γ acts faithfully on R ⇐⇒ ∃ left-inv’t order on Γ .
Conjecture
5 / 16
� left-inv’t order for Γ = large arith grp � SL(3, Z), etc.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
Park City, July 2012
∗


Spse ∃ left-inv’t order on SL(3, Z) = �
4
�
�
�
5
�
1 ,�
2 ,�
3 = Heisenberg group.
i.e., zn ≺ s, ∀n ∈ Z.
6 / 16

�
1 �
2

∗ �
3 
.
�
6
∗
There are actually 6 Heisenberg groups in Γ :
�
1 ,�
2 ,�
3 ,
�
2 ,�
3 ,�
4 ,
�
3 ,�
4 ,�
5
�
4 ,�
5 ,�
6 ,
�
5 ,�
6 ,�
1 ,
�
6 ,�
1 ,�
2 .
�
�
Replace x, y, z with inverse.)
−1
Interchange x and y: [y, x] = z .
Park City, July 2012
(break ties)
Exercise
≺ is a total order on Γ that is left-invariant.
(a ≺ b �⇒ ca ≺ cb) Hint: orient-pres: s x < y �⇒ c(x) < c(y).

z = x −1 y −1 xy
⇒ xy = yxz
2
⇒ x n y n = y n x n zn
(Recall z ∈ Z(H))
2
n n −n −n
⇒x y x y
= z−n .
(quadratic)
Suppose zp � x and zq � y.
Therefore e ≺ x −1 zp , y −1 zq , x, y
⇒ e ≺ x n y n (x −1 zp )n (y −1 zq )n
= x n y n x −n y −n zpn+qn
2
= z−n z(p+q)n = znegative . →←
3 / 16
Hint: (Γ , ≺) � (Q, <) �⇒ Dedekind completion of Γ is R.

Z

Z
1
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
(if Γ �⊂ SL(2, R))
Proof of Fact (⇐).
�
�
Γ � SL(3, Z) or Γ � SL 2, Z[α] or . . .
α = real, irrat alg’ic integer.
Γ � SO(1, n), SU(1, n), Sp(1, n), F4,1 .
Proof.
H has left-inv’t order.
(N, G/N left ord’ble ⇒ G left ord’ble
Γ ⊂ SO(1, 3) �⇒ Γ̇ acts on R
∃s ∈ {x ±1 , y ±1 }, z � s,
Exercise
z = [x, y] = x −1 y −1 xy ∈ Z(H).
(optional)
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
∀ left-ordering of H =
Notation
x=

homo φ : Γ → Homeo+ (R) ?
Lemma
Proposition (Witte, 1994)

Γ acts on S 1 �⇒ ∃ finite orbit
�⇒ Γ̇ has a fixed point.
1
Arith grps known to act on R are “small.”
But Homeo+ (R) has no elt’s of finite order:
ϕ(0) > 0 �⇒ ϕ2 (0) > ϕ(0) > 0 �⇒ ϕ3 (0) > 0
�⇒ . . . �⇒ ϕn (0) > 0.
(if Γ �⊂ SL(2, R))
Theorem (Ghys, Burger-Monod, Bader-Furman)
� SL(n, Z) or . . .
Example (Agol, Boyer-Rolfsen-Wiest)
�


1 Z Z


H = Heisenberg grp =  1 Z.
1
(faithful)
( Γ = arith grp � SL(n, Z) )
Γ acts on one-pt compactification of R ≈ S 1 .
Example
Γ � SL(2, Z) finite-index subgrp can be a free group.
Has many actions on R.
( Γ = arith grp )
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
Proof (⇒).
X = simplest possible space
= connected manifold of dim’n 1
= circle or line
¿∃
action of Γ on R or S 1 ?
Γ̇ acts on R ⇐⇒ Γ̈ acts on S 1
(faithful: no kernel)
Γ = arithmetic group
almost
(faithful)
Fact
In these lectures:
Abstract. The group SL(3, Z) cannot act (nontrivially) on the
circle (by homeomorphisms). We will see that many other
arithmetic groups also cannot act on the circle. The
discussion will involve several important topics in group
theory, such as amenability, Kazhdan’s property (T ), ordered
groups, bounded generation, and bounded cohomology.
Park City, July 2012
¿∃
In Geometric Group Theory (and elsewhere):
Study group Γ by looking at spaces it can act on.
X = Hn , CAT(0) cube cplx, Euclidean bldg, etc.
Dave Witte Morris
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
Question
Lecture 1: Introduction
8 / 16
�
1 ,�
2 ,�
3 = Heis grp ⇒ �
2 ��
1 or �
2 ��
3 .
Wolog �
2 ��
3 .
�
2 ,�
3 ,�
4 = Heis grp ⇒ �
3 ��
2 or �
3 ��
4 .
Must have �
3 ��
4 .
etc.
�
2 ��
3 ��
4 ��
5 ��
6 ��
1 ��
2
⇒�
2 ��
2 .
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
Park City, July 2012
9 / 16
Conjecture
Γ does not act on R if Γ = large arithmetic group.
Conjecture
Γ does not act on R (or S 1 ) if Γ = large arith group.
Proposition (Witte, 1994)
Γ does not act on R if Γ � SL(3, Z) or Sp(4, Z)
or contains either.
I.e., rankQ (Γ ) ≥ 2.
Remark
Large arithmetic groups usually have
Kazhdan’s Property (T ).
Remark
�
�
Proposition does not apply to SL 2, Z[α] .
Open problem
¿ Groups with Kazhdan’s Property (T ) have
˙ actions on R or S 1 ?
no
Open problem
Find arith group Γ , such that G/Γ is compact, and
finite-index subgroups of Γ does not act on R.
Theorem (Navas)
Groups with Kazhdan’s Property (T ) have
˙ C 2 actions on S 1 .
no
G/Γ compact �⇒ proposition never applies.
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
Park City, July 2012
10 / 16
Exercises
(faithfully) on R iff Γ is left-orderable.
(For ⇒, need to show that ties can be broken in a consistent way.)
2) In the Heisenberg group H, show:
a) z = [x, y] ∈ Z(H).
b) x k y � = y � x k z k� for k, � ∈ Z.
used in proof of Ghys
3) The proof that Γ = SL(3, Z) is not left-orderable:
�
� �
�
a) Verify: �
1 ,�
2 ,�
3 , �
2 ,�
3 ,�
4 , etc
are all isomorphic to the Heisenberg grp H.
b) Generalize proof to finite-index subgroups.
free group has a
faithful
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
action on R.
Park City, July 2012
13 / 16
Further reading
V. M. Kopytov and N. Ya. Medvedev:
Right-Ordered Groups. Consultants Bureau, New
York, 1996.
É. Ghys: Groups acting on the circle.
L’Enseignement Mathématique 47 (2001)
329–407. http://retro.seals.ch/cntmng;
?type=pdf&rid=ensmat-001:2001:47::210
A. Navas: Groups of Circle Diffeomorphisms,
Univ. of Chicago Press, 2011.
http://arxiv.org/abs/math/0607481
D. Morris: Introduction to Arithmetic Groups
(preprint). http://people.uleth.ca/~dave.
morris/books/IntroArithGroups.html
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
Park City, July 2012
16 / 16
(quasimorphisms)
(∃ finite orbit)
All lectures are essentially independent.
11 / 16
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
Park City, July 2012
12 / 16
Related reading
6) Torsion-free, nilpotent groups are left-ord’ble.
Some torsion-free, solvable groups are
not left-orderable!
(harder)
(Assumption: every finitely generated subgrp of Γ is left-ord’ble.)
9) Residually left-ord’ble �⇒ left-ord’ble.
(Assumption: ∀g ∈ Γ , ∃ homo ϕ : Γ → H, such that ϕ(g) ≠ e and
H is left-orderable.)
10) Locally indicable �⇒ left-orderable.
(Assumption: the abelianization of every nontrivial, finitely
generated subgroup is infinite.)
�
�
11) SL(3, Z) not isomorphic to subgrp of SL 2, Z[α]
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
(∃ finite orbit)
used in proof of Burger-Monod
5) Torsion-free, abelian groups are left-orderable.
7)
Coming up:
Fri: Intro to bounded cohomology
8) Locally left-orderable �⇒ left-orderable.
c) H is left-orderable.
fin gen
Park City, July 2012
Γ does not act on R (or S 1 ) if Γ = large arith group.
�
�
Tues: proof for SL 2, Z[α]
(and others)
bounded generation
Thurs: What is an amenable group?
Optional exercises
1) Show Γ acts
4) Every
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
Conjecture
Park City, July 2012
14 / 16
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
D. Witte: Arithmetic groups of higher Q-rank
cannot act on 1-manifolds, Proc. Amer. Math.
Soc. 122 (1994) 333–340.
http://www.jstor.org/stable/2161021
S. Boyer, D. Rolfsen, and B. Wiest: Orderable
3-manifold groups, Ann. Inst. Fourier (Grenoble)
55 (2005), no. 1, 243–288.
http://dx.doi.org/10.5802/aif.2098
Dave Witte Morris (Univ. of Lethbridge) Arithmetic grps do not act on S 1 (1/4)
Park City, July 2012
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