Some arithmetic groups that cannot act on the...

Some arithmetic groups thatcannot act on the line

Dave Witte Morris

University of Lethbridge, Alberta, Canadahttp://people.uleth.ca/�dave.morris

[email protected]

joint with

Lucy Lifschitz, University of Oklahoma

Vladimir Chernousov, University of Alberta

Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 1 / 16

Transformation groups

Given: group — ,

(connected)

manifold M .¿ What are the actions of — on M ?

I.e.: ¿ What are homos � : — ! Homeo��M� ?

Question¿ 9 action ?

Simplest casedimM � 1, so M � S1 or R.

Assume M � R.

ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)


Transformation groupsGiven:

group — ,

(connected)





Assume M � R.



Transformation groupsGiven: group — ,

(connected)





Assume M � R.



Transformation groupsGiven: group — ,

(connected)

manifold M .

¿ What are the actions of — on M ?I.e.: ¿ What are homos � : — ! Homeo��M� ?



Assume M � R.



Transformation groupsGiven: group — , (connected) manifold M .

¿ What are the actions of — on M ?I.e.: ¿ What are homos � : — ! Homeo��M� ?



Assume M � R.



Transformation groupsGiven: group — , (connected) manifold M .¿

What are the actions of — on M ?I.e.: ¿ What are homos � : — ! Homeo��M� ?



Assume M � R.



Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions

of — on M ?I.e.: ¿ What are homos � : — ! Homeo��M� ?



Assume M � R.



Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of —

on M ?I.e.: ¿ What are homos � : — ! Homeo��M� ?



Assume M � R.



Transformation groupsGiven: group — , (connected) manifold M .¿ What are the actions of — on M ?




Assume M � R.




I.e.:

¿ What are homos � : — ! Homeo��M� ?



Assume M � R.




I.e.: ¿ What are homos

� : — ! Homeo��M� ?



Assume M � R.




I.e.: ¿ What are homos � :

— ! Homeo��M� ?



Assume M � R.




I.e.: ¿ What are homos � : —

! Homeo��M� ?



Assume M � R.




I.e.: ¿ What are homos � : — !

Homeo��M� ?



Assume M � R.







Assume M � R.





Question

¿ 9 action ?


Assume M � R.





Question¿

9 action ?


Assume M � R.





Question¿ 9

action ?


Assume M � R.







Assume M � R.





Question¿ 9 (nontrivial) action ?


Assume M � R.





Question¿ 9 (faithful) action ?


Assume M � R.






Simplest case

dimM � 1, so M � S1 or R.Assume M � R.






Simplest casedimM �

1, so M � S1 or R.Assume M � R.






Simplest casedimM � 1,

so M � S1 or R.Assume M � R.






Simplest casedimM � 1, so

M � S1 or R.Assume M � R.






Simplest casedimM � 1, so M �

S1 or R.Assume M � R.






Simplest casedimM � 1, so M � S1

or R.Assume M � R.






Simplest casedimM � 1, so M � S1 or

R.Assume M � R.







Assume M � R.







Assume

M � R.







Assume M �

R.







Assume M � R.







Assume M � R.

Example

Z acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)






Assume M � R.

ExampleZ

acts on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)






Assume M � R.

ExampleZ acts

on R. (Tn�x� � x�n =) Tm�n � Tm �Tn)






Assume M � R.

ExampleZ acts on

R. (Tn�x� � x�n =) Tm�n � Tm �Tn)






Assume M � R.

ExampleZ acts on R.

(Tn�x� � x�n =) Tm�n � Tm �Tn)






Assume M � R.

ExampleZ acts on R. (Tn�x�

� x�n =) Tm�n � Tm �Tn)






Assume M � R.

ExampleZ acts on R. (Tn�x� �

x�n =) Tm�n � Tm �Tn)






Assume M � R.

ExampleZ acts on R. (Tn�x� � x

�n =) Tm�n � Tm �Tn)






Assume M � R.

ExampleZ acts on R. (Tn�x� � x�

n =) Tm�n � Tm �Tn)






Assume M � R.

ExampleZ acts on R. (Tn�x� � x�n

=) Tm�n � Tm �Tn)






Assume M � R.

ExampleZ acts on R. (Tn�x� � x�n =)

Tm�n � Tm �Tn)






Assume M � R.

ExampleZ acts on R. (Tn�x� � x�n =) Tm�n

� Tm �Tn)






Assume M � R.

ExampleZ acts on R. (Tn�x� � x�n =) Tm�n �

Tm �Tn)






Assume M � R.

ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm

�Tn)






Assume M � R.

ExampleZ acts on R. (Tn�x� � x�n =) Tm�n � Tm �

Tn)






Assume M � R.







Assume M � R.



Question¿ 9 (faithful) action of — on R ?

Today: — is an arithmetic group

ExampleSL�2;Z� does not act on R.

Proof."�1 00 �1

#2� I. So SL�2;Z� has elt’s of finite order.

But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0

=) . . . =) ’n�0� > 0.



Today:

— is an arithmetic group


Proof."�1 00 �1



=) . . . =) ’n�0� > 0.



Today: — is

an arithmetic group


Proof."�1 00 �1



=) . . . =) ’n�0� > 0.





Proof."�1 00 �1



=) . . . =) ’n�0� > 0.




Example

SL�2;Z� does not act on R.

Proof."�1 00 �1



=) . . . =) ’n�0� > 0.




ExampleSL�2;Z�

does not act on R.

Proof."�1 00 �1



=) . . . =) ’n�0� > 0.




ExampleSL�2;Z� does not

act on R.

Proof."�1 00 �1



=) . . . =) ’n�0� > 0.





Proof."�1 00 �1



=) . . . =) ’n�0� > 0.





Proof.

"�1 00 �1



=) . . . =) ’n�0� > 0.





Proof."�1 00 �1

#

2� I. So SL�2;Z� has elt’s of finite order.


=) . . . =) ’n�0� > 0.





Proof."�1 00 �1

#2

� I. So SL�2;Z� has elt’s of finite order.


=) . . . =) ’n�0� > 0.





Proof."�1 00 �1

#2� I.

So SL�2;Z� has elt’s of finite order.


=) . . . =) ’n�0� > 0.





Proof."�1 00 �1

#2� I. So

SL�2;Z� has elt’s of finite order.


=) . . . =) ’n�0� > 0.





Proof."�1 00 �1

#2� I. So SL�2;Z� has

elt’s of finite order.


=) . . . =) ’n�0� > 0.





Proof."�1 00 �1



=) . . . =) ’n�0� > 0.





Proof."�1 00 �1


But

Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0

=) . . . =) ’n�0� > 0.





Proof."�1 00 �1


But Homeo��R� has

no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0

=) . . . =) ’n�0� > 0.





Proof."�1 00 �1


But Homeo��R� has no elt’s of finite order:

’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.





Proof."�1 00 �1


But Homeo��R� has no elt’s of finite order:’

�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.





Proof."�1 00 �1


But Homeo��R� has no elt’s of finite order:’�0�

> 0 =) ’2�0� > ’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.





Proof."�1 00 �1


But Homeo��R� has no elt’s of finite order:’�0� > 0

=) ’2�0� > ’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.





Proof."�1 00 �1


But Homeo��R� has no elt’s of finite order:’�0� > 0 =)

’2�0� > ’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.





Proof."�1 00 �1


But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0�

> ’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.





Proof."�1 00 �1


But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� >

’�0� > 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.





Proof."�1 00 �1


But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0�

> 0 =) ’3�0� > 0=) . . . =) ’n�0� > 0.





Proof."�1 00 �1


But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0

=) ’3�0� > 0=) . . . =) ’n�0� > 0.





Proof."�1 00 �1


But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =)

’3�0� > 0=) . . . =) ’n�0� > 0.





Proof."�1 00 �1


But Homeo��R� has no elt’s of finite order:’�0� > 0 =) ’2�0� > ’�0� > 0 =) ’3�0�

> 0=) . . . =) ’n�0� > 0.





Proof."�1 00 �1



=) . . . =) ’n�0� > 0.





Proof."�1 00 �1



=)

. . . =) ’n�0� > 0.





Proof."�1 00 �1



=) . . .

=) ’n�0� > 0.





Proof."�1 00 �1



=) . . . =)

’n�0� > 0.





Proof."�1 00 �1



=) . . . =) ’n�0�

> 0.





Proof."�1 00 �1



=) . . . =) ’n�0� > 0.





Proof."�1 00 �1



=) . . . =) ’n�0� > 0.


Example:

SL�2;Z� does not act on Rbecause it has elements of finite order.

Example— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.

FactThere exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).

ConjectureLarge arithmetic groups (R-rank > 1) cannot act on R.


Example: SL�2;Z�

does not act on Rbecause it has elements of finite order.





Example: SL�2;Z� does not act on R

because it has elements of finite order.





Example: SL�2;Z� does not act on Rbecause

it has elements of finite order.





Example: SL�2;Z� does not act on Rbecause it has elements of finite order.






Example

— É SL�2;Z� finite-index subgrp can be a free group.Has many actions on R.





Example— É SL�2;Z�

finite-index subgrp can be a free group.Has many actions on R.





Example— É SL�2;Z� finite-index subgrp

can be a free group.Has many actions on R.





Example— É SL�2;Z� finite-index subgrp can be

a free group.Has many actions on R.





Example— É SL�2;Z� finite-index subgrp can be a free group.

Has many actions on R.





Example— É SL�2;Z� finite-index subgrp can be a free group.Has

many actions on R.











Fact

There exist other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).





FactThere exist

other examples that act on R.But all are “small”. (I think all known are in SO�1; n�).





FactThere exist other examples

that act on R.But all are “small”. (I think all known are in SO�1; n�).





FactThere exist other examples that act on R.

But all are “small”. (I think all known are in SO�1; n�).





FactThere exist other examples that act on R.But all are

“small”. (I think all known are in SO�1; n�).





FactThere exist other examples that act on R.But all are “small”.

(I think all known are in SO�1; n�).





FactThere exist other examples that act on R.But all are “small”. (I think

all known are in SO�1; n�).





FactThere exist other examples that act on R.But all are “small”. (I think all known

are in SO�1; n�).





FactThere exist other examples that act on R.But all are “small”. (I think all known are in

SO�1; n�).











Conjecture

Large arithmetic groups (R-rank > 1) cannot act on R.





ConjectureLarge arithmetic groups

(R-rank > 1) cannot act on R.





ConjectureLarge arithmetic groups (R-rank

> 1) cannot act on R.





ConjectureLarge arithmetic groups (R-rank > 1)

cannot act on R.












Question¿ 9 (nontrivial) action of — on R ?

Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)

Or — É SL�2;Z�

p3��

or . . .Or — É SL

�2;Z��

�� = real, irrat alg’ic integer.

But — � SL�2;Z�, other “small" grps. (Need rankR — > 1.)

Conjecture— does not act on R.



Assume

— is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)


p3��


�2;Z��






Assume — is a

“large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)


p3��


�2;Z��






Assume — is a “large”

arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)


p3��


�2;Z��






Assume — is a “large” arithmetic group:

— É SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)


p3��


�2;Z��






Assume — is a “large” arithmetic group:— É

SL�3;Z� = f 3� 3 integer matrices of det 1 g(subgroup of finite index)


p3��


�2;Z��






Assume — is a “large” arithmetic group:— É SL�3;Z�

= f 3� 3 integer matrices of det 1 g(subgroup of finite index)


p3��


�2;Z��






Assume — is a “large” arithmetic group:— É SL�3;Z� = f 3� 3 integer matrices of det 1 g

(subgroup of finite index)


p3��


�2;Z��








p3��


�2;Z��







Or

— É SL�2;Z�

p3��


�2;Z��







Or — É

SL�2;Z�

p3��


�2;Z��







Or — É SL�2;Z

�p

3��


�2;Z��








p3��


�2;Z��








p3��

or

. . .Or — É SL

�2;Z��








p3��

or . . .

Or — É SL�2;Z��








p3��

or . . .Or

— É SL�2;Z��








p3��


�2;Z

��

� = real, irrat alg’ic integer.







p3��


�2;Z��

�

� = real, irrat alg’ic integer.







p3��


�2;Z��

�� =

real, irrat alg’ic integer.







p3��


�2;Z��

�� = real,

irrat alg’ic integer.







p3��


�2;Z��

�� = real, irrat

alg’ic integer.







p3��


�2;Z��








p3��


�2;Z��


But

— � SL�2;Z�, other “small" grps. (Need rankR — > 1.)






p3��


�2;Z��


But — �

SL�2;Z�, other “small" grps. (Need rankR — > 1.)






p3��


�2;Z��


But — � SL�2;Z�,

other “small" grps. (Need rankR — > 1.)






p3��


�2;Z��


But — � SL�2;Z�, other

“small" grps. (Need rankR — > 1.)






p3��


�2;Z��


But — � SL�2;Z�, other “small"

grps. (Need rankR — > 1.)






p3��


�2;Z��


But — � SL�2;Z�, other “small" grps.

(Need rankR — > 1.)






p3��


�2;Z��


But — � SL�2;Z�, other “small" grps. (Need

rankR — > 1.)






p3��


�2;Z��


But — � SL�2;Z�, other “small" grps. (Need rankR —

> 1.)






p3��


�2;Z��








p3��


�2;Z��



Conjecture

— does not act on R.





p3��


�2;Z��



Conjecture— does not

act on R.





p3��


�2;Z��








p3��


�2;Z��





Conjecture— no action on R if — É SL�3;Z� or SL�2;Z��

�or . . .

Theorem (Witte, Lifschitz-Morris)— no action on R if — É SL�3;Z� or SL�2;Z��

�or . . .

Proof combines bdd generation and bdd orbits.

Unipotent subgroups: U �"

1 �0 1

#, V �

"1 0� 1

#.

Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V

boundedly

generate — (up to finite index).— acts on R =) U -orbits

(and V -orbits)

are bdd.



�or . . .



1 �0 1

#, V �

"1 0� 1

#.


boundedly


(and V -orbits)

are bdd.



�



1 �0 1

#, V �

"1 0� 1

#.


boundedly


(and V -orbits)

are bdd.



�Proof

combines bdd generation and bdd orbits.


1 �0 1

#, V �

"1 0� 1

#.


boundedly


(and V -orbits)

are bdd.



�Proof combines

bdd generation and bdd orbits.


1 �0 1

#, V �

"1 0� 1

#.


boundedly


(and V -orbits)

are bdd.



�Proof combines bdd generation

and bdd orbits.


1 �0 1

#, V �

"1 0� 1

#.


boundedly


(and V -orbits)

are bdd.



�Proof combines bdd generation and

bdd orbits.


1 �0 1

#, V �

"1 0� 1

#.


boundedly


(and V -orbits)

are bdd.



�Proof combines bdd generation and bdd orbits.


1 �0 1

#, V �

"1 0� 1

#.


boundedly


(and V -orbits)

are bdd.




Unipotent subgroups:

U �"

1 �0 1

#, V �

"1 0� 1

#.


boundedly


(and V -orbits)

are bdd.




Unipotent subgroups: U �

"1 �0 1

#, V �

"1 0� 1

#.


boundedly


(and V -orbits)

are bdd.





1 �0 1

#,

V �"

1 0� 1

#.


boundedly


(and V -orbits)

are bdd.





1 �0 1

#, V �

"1 0� 1

#.


boundedly


(and V -orbits)

are bdd.





1 �0 1

#, V �

"1 0� 1

#.


boundedly


(and V -orbits)

are bdd.





1 �0 1

#, V �

"1 0� 1

#.

Theorem (Carter-Keller-Paige, Lifschitz-Morris)

U and V

boundedly


(and V -orbits)

are bdd.





1 �0 1

#, V �

"1 0� 1

#.


boundedly


(and V -orbits)

are bdd.





1 �0 1

#, V �

"1 0� 1

#.


boundedly

generate —

(up to finite index).— acts on R =) U -orbits

(and V -orbits)

are bdd.





1 �0 1

#, V �

"1 0� 1

#.


boundedly

generate — (up to finite index).

— acts on R =) U -orbits

(and V -orbits)

are bdd.





1 �0 1

#, V �

"1 0� 1

#.

Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).

— acts on R =) U -orbits

(and V -orbits)

are bdd.





1 �0 1

#, V �

"1 0� 1

#.

Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).— acts on R

=) U -orbits

(and V -orbits)

are bdd.





1 �0 1

#, V �

"1 0� 1

#.

Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).— acts on R =)

U -orbits

(and V -orbits)

are bdd.





1 �0 1

#, V �

"1 0� 1

#.

Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).— acts on R =) U -orbits

(and V -orbits)

are bdd.





1 �0 1

#, V �

"1 0� 1

#.

Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).— acts on R =) U -orbits

(and V -orbits)

are bdd.





1 �0 1

#, V �

"1 0� 1

#.

Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).— acts on R =) U -orbits (and V -orbits) are bdd.





1 �0 1

#, V �

"1 0� 1

#.

Theorem (Carter-Keller-Paige, Lifschitz-Morris)U and V boundedly generate — (up to finite index).— acts on R =) U -orbits (and V -orbits) are bdd.


Bounded generation by unip subgrps

Note: Invertible matrix � Id by row operations.

Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops.

Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.

But # steps is not bounded:U and V do not boundedly generate SL

�2;Z�.



Note:

Invertible matrix � Id by row operations.


Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.



Note: Invertible matrix

� Id by row operations.


Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.



Note: Invertible matrix �

Id by row operations.


Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.



Note: Invertible matrix � Id

by row operations.


Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.



Note: Invertible matrix � Id by

row operations.


Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.




Key fact:

g 2 SL�2;Z� � Id by integer (Z) row ops.

Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.




Key fact: g 2 SL�2;Z�

� Id by integer (Z) row ops.

Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.




Key fact: g 2 SL�2;Z� �

Id by integer (Z) row ops.

Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.




Key fact: g 2 SL�2;Z� � Id

by integer (Z) row ops.

Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.




Key fact: g 2 SL�2;Z� � Id by

integer (Z) row ops.

Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.




Key fact: g 2 SL�2;Z� � Id by integer

(Z) row ops.

Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.




Key fact: g 2 SL�2;Z� � Id by integer (Z)

row ops.

Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example

"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#

�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#

�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#

�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#

�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.

But

# steps is not bounded:U and V do not boundedly generate SL

�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.

But # steps

is not bounded:U and V do not boundedly generate SL

�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.

But # steps is

not bounded:U and V do not boundedly generate SL

�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.

But # steps is not bounded:

U and V do not boundedly generate SL�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.

But # steps is not bounded:U and V

do not boundedly generate SL�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.

But # steps is not bounded:U and V do not

boundedly generate SL�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.

But # steps is not bounded:U and V do not boundedly generate

SL�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.





Example"13 315 12

#�

"3 75 12

#�

"3 72 5

#

�

"1 22 5

#�

"1 20 1

#�

"1 00 1

#.


�2;Z�.


Key fact:

g 2 SL�2;Z� � Id by integer (Z) row ops,

but # steps is not bounded.

Remark: In SL�3;Z�, # steps is bounded [Carter-Keller].

Theorem (Liehl, Carter-Keller-Paige)For Z�� row operations, # steps is bounded.

9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.

I.e., U and V boundedly gen — � SL�2;Z��

�.

So SL�2;Z��

�= UVUV � � �UV .


Key fact: g 2 SL�2;Z� � Id

by integer (Z) row ops,but # steps is not bounded.



9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .


Key fact: g 2 SL�2;Z� � Id by integer (Z) row ops,




9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .




Remark:

In SL�3;Z�, # steps is bounded [Carter-Keller].


9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .




Remark: In SL�3;Z�,

# steps is bounded [Carter-Keller].


9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .




Remark: In SL�3;Z�, # steps is bounded

[Carter-Keller].


9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .





Theorem (Liehl, Carter-Keller-Paige)

For Z�� row operations, # steps is bounded.9n, 8g 2 SL

�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .





Theorem (Liehl, Carter-Keller-Paige)For

Z�� row operations, # steps is bounded.9n, 8g 2 SL

�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .





Theorem (Liehl, Carter-Keller-Paige)For Z��

row operations, # steps is bounded.9n, 8g 2 SL

�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .





Theorem (Liehl, Carter-Keller-Paige)For Z�� row operations,

# steps is bounded.9n, 8g 2 SL

�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .






9n,

8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2

SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z

��, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�,

g � u1v1u2v2� � �unvn.I.e., U and V boundedly gen — � SL

�2;Z��

�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g

� u1v1u2v2� � �unvn.I.e., U and V boundedly gen — � SL

�2;Z��

�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g �

u1v1u2v2� � �unvn.I.e., U and V boundedly gen — � SL

�2;Z��

�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1

v1u2v2� � �unvn.I.e., U and V boundedly gen — � SL

�2;Z��

�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1

u2v2� � �unvn.I.e., U and V boundedly gen — � SL

�2;Z��

�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2

v2� � �unvn.I.e., U and V boundedly gen — � SL

�2;Z��

�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2

� � �unvn.I.e., U and V boundedly gen — � SL

�2;Z��

�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �

unvn.I.e., U and V boundedly gen — � SL

�2;Z��

�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �un

vn.I.e., U and V boundedly gen — � SL

�2;Z��

�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.

I.e.,

U and V boundedly gen — � SL�2;Z��

�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.

I.e., U and V

boundedly gen — � SL�2;Z��

�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.

I.e., U and V boundedly gen

— � SL�2;Z��

�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So

SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�

= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�=

UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .






9n, 8g 2 SL�2;Z��

�, g � u1v1u2v2� � �unvn.


�.

So SL�2;Z��

�= UVUV � � �UV .


Theorem (Liehl)

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 proofAssume Artin’s Conjecture:8r � �1, perfect square,

9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q

�

f1;2;3; : : : ; q� 1gAssume 9q in every arith progression fa� kbg.

9 q � a� kb,

p

is a primitive root modulo q.


Theorem (Liehl)SL�2;Z�1=p�

�

bddly gen’d by elem mats.I.e., T � Id by Z�1=p� col ops, # steps is bdd.



�


9 q � a� kb,

p




�bddly gen’d

by elem mats.I.e., T � Id by Z�1=p� col ops, # steps is bdd.



�


9 q � a� kb,

p








�


9 q � a� kb,

p





I.e.,

T � Id by Z�1=p� col ops, # steps is bdd.



�


9 q � a� kb,

p





I.e., T � Id

by Z�1=p� col ops, # steps is bdd.



�


9 q � a� kb,

p





I.e., T � Id by

Z�1=p� col ops, # steps is bdd.



�


9 q � a� kb,

p





I.e., T � Id by Z�1=p� col ops,

# steps is bdd.



�


9 q � a� kb,

p





I.e., T � Id by Z�1=p� col ops, # steps

is bdd.



�


9 q � a� kb,

p








�


9 q � a� kb,

p






Easy proof

Assume Artin’s Conjecture:8r � �1, perfect square,


�


9 q � a� kb,

p






Easy proofAssume

Artin’s Conjecture:8r � �1, perfect square,


�


9 q � a� kb,

p






Easy proofAssume Artin’s Conjecture:

8r � �1, perfect square,9 1 primes q, s.t. r is primitive root modulo q:

f r ; r 2; r 3; : : : g mod q

�


9 q � a� kb,

p






Easy proofAssume Artin’s Conjecture:8r

� �1, perfect square,9 1 primes q, s.t. r is primitive root modulo q:

f r ; r 2; r 3; : : : g mod q

�


9 q � a� kb,

p






Easy proofAssume Artin’s Conjecture:8r � �1,

perfect square,9 1 primes q, s.t. r is primitive root modulo q:

f r ; r 2; r 3; : : : g mod q

�


9 q � a� kb,

p








�


9 q � a� kb,

p







9

1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q

�


9 q � a� kb,

p







9 1 primes q,

s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q

�


9 q � a� kb,

p







9 1 primes q, s.t.

r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q

�


9 q � a� kb,

p







9 1 primes q, s.t. r is primitive root

modulo q:f r ; r 2; r 3; : : : g mod q

�


9 q � a� kb,

p







9 1 primes q, s.t. r is primitive root modulo q:

f r ; r 2; r 3; : : : g mod q

�


9 q � a� kb,

p







9 1 primes q, s.t. r is primitive root modulo q:f

r ; r 2; r 3; : : : g mod q

�


9 q � a� kb,

p







9 1 primes q, s.t. r is primitive root modulo q:f r ;

r 2; r 3; : : : g mod q

�


9 q � a� kb,

p







9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2;

r 3; : : : g mod q

�


9 q � a� kb,

p







9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3;

: : : g mod q

�


9 q � a� kb,

p







9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g

mod q

�


9 q � a� kb,

p








�


9 q � a� kb,

p








�

f1;2;3; : : : ; q� 1g

Assume 9q in every arith progression fa� kbg.

9 q � a� kb,

p







9 1 primes q, s.t. r is primitive root modulo q:f r ; r 2; r 3; : : : g mod q � f1;2;3; : : : ; q� 1g


9 q � a� kb,

p








Assume

9q in every arith progression fa� kbg.

9 q � a� kb,

p








Assume 9q

in every arith progression fa� kbg.

9 q � a� kb,

p








Assume 9q in

every arith progression fa� kbg.

9 q � a� kb,

p








Assume 9q in every arith progression

fa� kbg.

9 q � a� kb,

p









9 q � a� kb,

p









9 q

� a� kb,

p









9 q � a� kb,

p









9 q � a� kb,

p is a primitive root modulo q.








9 q � a� kb,

p









9 q � a� kb, p is a primitive root modulo q.








9 q � a� kb, p is a primitive root modulo q.





Proof."a bc d

#q � a� kb prime, p is prim root

�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �

#p‘ unit: can add anything to q

�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof.

"a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d

#

q � a� kb prime, p is prim root

�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d

#q � a� kb prime,

p is prim root

�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#

p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ �

b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�;

p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b

� k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �

#

p‘ unit: can add anything to q

�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �

#p‘ unit:

can add anything to q

�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �

#p‘ unit: can add

anything to q

�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �

#p‘ unit: can add anything

to q

�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#

�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#

�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.





Proof."a bc d


�

"q b� �

#p‘ � b �mod q�; p‘ � b � k0q

�

"q p‘� �


�

"1 p‘� �

#�

"1 0� 1

#�

"1 00 1

#.


Bdd generation:

— � UVUV � � �UV .

Bdd orbits: U -orbits and V -orbits are bounded.

Corollary� : — ! Homeo��R�

) every — -orbit on R is bounded) — has a fixed point.

Corollary— cannot act on R.


Bdd generation: — �

UVUV � � �UV .






Bdd generation: — � UVUV � � �UV .







Bdd orbits:

U -orbits and V -orbits are bounded.






Bdd orbits: U -orbits

and V -orbits are bounded.






Bdd orbits: U -orbits and V -orbits

are bounded.













Corollary

� : — ! Homeo��R�) every — -orbit on R is bounded) — has a fixed point.












)

every — -orbit on R is bounded) — has a fixed point.






) every — -orbit

on R is bounded) — has a fixed point.






) every — -orbit on R

is bounded) — has a fixed point.






) every — -orbit on R is bounded

) — has a fixed point.






) every — -orbit on R is bounded)

— has a fixed point.






) every — -orbit on R is bounded) — has

a fixed point.













Corollary

— cannot act on R.















Proof.Suppose there is a nontrivial action.

It has fixed points:

Remove them:

Take a connected component:

— acts on open interval (� R) with no fixed point.!



Proof.

Suppose there is a nontrivial action.


Remove them:







Remove them:







Remove them:






It has

fixed points:

Remove them:







Remove them:







Remove them:







Remove them:







Remove them:







Remove them:

Take

a connected component:






Remove them:







Remove them:







Remove them:


— acts on

open interval (� R) with no fixed point.!





Remove them:


— acts on open interval

(� R) with no fixed point.!





Remove them:


— acts on open interval (�

R) with no fixed point.!





Remove them:


— acts on open interval (� R)

with no fixed point.!





Remove them:


— acts on open interval (� R) with

no fixed point.!





Remove them:


— acts on open interval (� R) with no fixed point.

!





Remove them:







Remove them:







Remove them:




Bounded orbits

Theorem (Lifschitz-Morris)— � SL�2;Z�1=p�

�acts on R ) every U -orbit bdd.

u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit

of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)

Wolog u�x� < v�x�.Then pn

�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits

Theorem (Lifschitz-Morris)

— � SL�2;Z�1=p�


u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits

Theorem (Lifschitz-Morris)— �

SL�2;Z�1=p�


u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits

Theorem (Lifschitz-Morris)— � SL�2;Z�

1=p��

acts on R ) every U -orbit bdd.

u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits


�

acts on R ) every U -orbit bdd.

u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits


�acts on R

) every U -orbit bdd.

u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits


�acts on R )

every U -orbit bdd.

u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits


�acts on R ) every U -orbit

bdd.

u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �

"1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#,

v �"

1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#,

p �"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#

Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume

U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit

of x

not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)


�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit

of x not bdd

above.Assume p fixes x. (p does have fixed pts, so not an issue.)


�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit

and V -orbit

of x not bdd above.

Assume p fixes x. (p does have fixed pts, so not an issue.)


�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit and V -orbit of x not bdd above.

Assume p fixes x. (p does have fixed pts, so not an issue.)


�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit and V -orbit of x not bdd above.Assume

p fixes x. (p does have fixed pts, so not an issue.)


�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit and V -orbit of x not bdd above.Assume p

fixes x. (p does have fixed pts, so not an issue.)


�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x.

(p does have fixed pts, so not an issue.)


�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts,

so not an issue.)


�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p

#Assume U -orbit and V -orbit of x not bdd above.Assume p fixes x. (p does have fixed pts, so not an issue.)


�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p


Wolog

u�x� < v�x�.Then pn

�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p


Wolog u�x� < v�x�.

Then pn�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p


Wolog u�x� < v�x�.Then

pn�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS

= pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS =

pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�

� �pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

��

�pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�

!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!

1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�

!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!

1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS

= pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS =

pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�

� �pnvp�n��x�! 0�x� <1.!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

��

�pnvp�n��x�! 0�x� <1.!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�

! 0�x� <1.!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�!

0�x� <1.!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x�

<1.!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <

1.!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

!


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.

! Dave Witte Morris (Univ. of Lethbridge) Arithmetic groups cannot act on R UVA (April 2010) 13 / 16

Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.


Bounded orbits



u �"

1 u0 1

#, v �

"1 0v 1

#, p �

"p 00 1=p



�u�x�

�< pn

�v�x�

�.

LHS = pn�u�x�

�� pnup�n��x�!1�x�!1.

RHS = pn�v�x�

�� pnvp�n��x�! 0�x� <1.


Other arithmetic groups of higher rank

PropositionSuppose —1 � —2.

If —2 acts on R, then —1 acts on R.

If —1 does not act on R, then —2 does not act on R.

Our methods require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.

Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then — :

� SL�2;Z��

�or noncocpct arith grp in SL�3;R� or SL�3;C�.



Proposition

Suppose —1 � —2.





� SL�2;Z��




PropositionSuppose

—1 � —2.





� SL�2;Z��









� SL�2;Z��





If

—2 acts on R, then —1 acts on R.




� SL�2;Z��





If —2

acts on R, then —1 acts on R.




� SL�2;Z��





If —2 acts on R,

then —1 acts on R.




� SL�2;Z��





If —2 acts on R, then

—1 acts on R.




� SL�2;Z��





If —2 acts on R, then —1

acts on R.




� SL�2;Z��









� SL�2;Z��






If

—1 does not act on R, then —2 does not act on R.



� SL�2;Z��






If —1

does not act on R, then —2 does not act on R.



� SL�2;Z��






If —1 does not

act on R, then —2 does not act on R.



� SL�2;Z��






If —1 does not act on R,

then —2 does not act on R.



� SL�2;Z��






If —1 does not act on R, then

—2 does not act on R.



� SL�2;Z��






If —1 does not act on R, then —2

does not act on R.



� SL�2;Z��









� SL�2;Z��







Our methods

require — to have a unipotent subgrp.Such arithmetic groups are called noncocompact.


� SL�2;Z��







Our methods require —

to have a unipotent subgrp.Such arithmetic groups are called noncocompact.


� SL�2;Z��







Our methods require — to have

a unipotent subgrp.Such arithmetic groups are called noncocompact.


� SL�2;Z��







Our methods require — to have a unipotent subgrp.

Such arithmetic groups are called noncocompact.


� SL�2;Z��







Our methods require — to have a unipotent subgrp.Such arithmetic groups

are called noncocompact.


� SL�2;Z��







Our methods require — to have a unipotent subgrp.Such arithmetic groups are called

noncocompact.


� SL�2;Z��









� SL�2;Z��








Theorem (Chernousov-Lifschitz-Morris)

Spse — is a noncocompact arith group of higher rank.Then — :

� SL�2;Z��








Theorem (Chernousov-Lifschitz-Morris)Spse

— is a noncocompact arith group of higher rank.Then — :

� SL�2;Z��








Theorem (Chernousov-Lifschitz-Morris)Spse — is

a noncocompact arith group of higher rank.Then — :

� SL�2;Z��








Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact

arith group of higher rank.Then — :

� SL�2;Z��








Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group

of higher rank.Then — :

� SL�2;Z��








Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.

Then — :� SL

�2;Z��








Theorem (Chernousov-Lifschitz-Morris)Spse — is a noncocompact arith group of higher rank.Then

— :� SL

�2;Z��









�

SL�2;Z��









� SL�2;Z��

�

or noncocpct arith grp in SL�3;R� or SL�3;C�.








� SL�2;Z��

�or

noncocpct arith grp in SL�3;R� or SL�3;C�.








� SL�2;Z��

�or noncocpct arith grp

in SL�3;R� or SL�3;C�.








� SL�2;Z��

�or noncocpct arith grp in

SL�3;R� or SL�3;C�.








� SL�2;Z��

�or noncocpct arith grp in SL�3;R�

or SL�3;C�.








� SL�2;Z��

�or noncocpct arith grp in SL�3;R� or

SL�3;C�.








� SL�2;Z��









� SL�2;Z��



Open Problem

Show noncocpct arith grps in SL�3;R� and SL�3;C�cannot act on R.

Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated by unips.

Rapinchuk Conjecture implies no action on Rif — noncocompact of higher rank.

Cocompact case will require new ideas.


Open ProblemShow noncocpct arith grps in SL�3;R� and SL�3;C�

cannot act on R.






cannot act on R.






cannot act on R.

Conjecture (Rapinchuk, �1990)

These arith grps are boundedly generated by unips.





cannot act on R.

Conjecture (Rapinchuk, �1990)These arith grps

are boundedly generated by unips.





cannot act on R.

Conjecture (Rapinchuk, �1990)These arith grps are

boundedly generated by unips.





cannot act on R.

Conjecture (Rapinchuk, �1990)These arith grps are boundedly generated

by unips.





cannot act on R.






cannot act on R.


Rapinchuk Conjecture

implies no action on Rif — noncocompact of higher rank.




cannot act on R.


Rapinchuk Conjecture implies

no action on Rif — noncocompact of higher rank.




cannot act on R.


Rapinchuk Conjecture implies no action on R

if — noncocompact of higher rank.




cannot act on R.


Rapinchuk Conjecture implies no action on Rif

— noncocompact of higher rank.




cannot act on R.


Rapinchuk Conjecture implies no action on Rif — noncocompact

of higher rank.




cannot act on R.






cannot act on R.



Cocompact case

will require new ideas.



cannot act on R.



Cocompact case will require

new ideas.



cannot act on R.






cannot act on R.





V. Chernousov, L. Lifschitz, and D. W. Morris:Almost-minimal nonuniform lattices of higher rank,Michigan Mathematical Journal 56, no. 2, (2008), 453Ð478.http://arxiv.org/abs/0705.4330

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

D. W. Morris:Bounded generation of SL�n;A� (after D. Carter, G. Keller and E. Paige),New York Journal of Mathematics 13 (2007) 383–421.http://nyjm.albany.edu/j/2007/13-17.html

A. Ondrus:Minimal anisotropic groups of higher real rank,(preprint, 2009, University of Alberta).

D. Witte:Arithmetic groups of higher Q-rank cannot act on 1-manifolds,Proc. Amer. Math. Soc. 122 (1994) 333–340.


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