Embed Size (px)
Citation preview
Symmetry and self-similarity in geometry
Wouter van Limbeek University of Michigan
University of Cambridge 29 January 2018
T 2 =
0
1
1
An example
T 2 =
0
1
11/n
1/n
An example
T 2 =
0
1
11/n
1/n
An example
T 2 =
0
1
11/n
1/n
An example
An example
T 2 =
T 2 =
0
1
11/n
1/n
n2 : 1
An example
T 2
T 2
0
1
11/n
1/n
n2 : 1
1. 9 covers with degree > 1
2. Symmetry covers
Remarks:
Genus 2�
8g � 2 : @ covers ⌃g ! ⌃g
� = 2� 2g 6= 0
Hurwitz’s 84(g � 1) Theorem (1893):
Reason:
1.
2.
⌃g Riemann surface, g � 2 =) |Aut+(⌃g)| 84(g � 1)
Genus 2�
Hurwitz’s 84(g � 1) Theorem (1893):
|Aut+(⌃g)| 84(g � 1)
Genus 2�
Hurwitz’s 84(g � 1) Theorem (1893):
|Aut+(⌃g)| 84(g � 1)
8X = H2/� : Area(X) � ⇡21
Key fact [ area estimate for
“orbifold” quotients
:[ #
(Image: Claudio Rocchini)
Genus 2�
Hurwitz’s 84(g � 1) Theorem (1893):
|Aut+(⌃g)| 84(g � 1)
8X = H2/� : Area(X) � ⇡21
Key fact [ area estimate for
“orbifold” quotients
:[
(Image: Claudio Rocchini)
Area(⌃g/Aut+(⌃g)) =Area(⌃g)
|Aut+(⌃g)|
� ⇡/21
Connection:
#
Two basic problems
9µ > 0 : 8� : vol(X/�) � µ ?
1.
#0
1
11/n
1/n
n2 : 1
Classify M that
self-cover with deg > 1.
Which Riem. mnfds X
have “minimal quotients”:
2.
Problem: Classify M that self-cover with deg > 1.
Problem:
dim = 2 : T 2,K
Low dimensions:
Classify M that self-cover with deg > 1.
Problem:
dim = 2 : T 2,K
Low dimensions:
dim = 3 [Yu-Wang ’99] : ⌃⇥ S1,T 2 ! M!
S1
Classify M that self-cover with deg > 1.
Problem:
dim = 2 : T 2,K
Low dimensions:
dim = 3 [Yu-Wang ’99] : ⌃⇥ S1,T 2 ! M!
S1
Kahler and dimC = 2, 3 [Horing-Peternell, ’11]
Classify M that self-cover with deg > 1.
Problem:
dim = 2 : T 2,K
Low dimensions:
dim = 3 [Yu-Wang ’99] : ⌃⇥ S1,T 2 ! M!
S1
Kahler and dimC = 2, 3 [Horing-Peternell, ’11]
dim � 4
1. Tori Tn= Rn/Zn
�A 2 Mn(Z), deg = | det(A)|
Examples
( )
Classify M that self-cover with deg > 1.
Problem:
dim = 2 : T 2,K
Low dimensions:
dim = 3 [Yu-Wang ’99] : ⌃⇥ S1,T 2 ! M!
S1
Kahler and dimC = 2, 3 [Horing-Peternell, ’11]
dim � 4
1. Tori Tn= Rn/Zn
�A 2 Mn(Z), deg = | det(A)|
Examples
( )
Nilmanifolds
2.
Classify M that self-cover with deg > 1.
Problem:
Examples
:Nilmanifolds
Classify M that self-cover with deg > 1.
Problem:
Examples
:
Gromov’s Expanding Maps Theorem [’81]:
f : M ! M expanding self-cover
(up to finite cover)
=) M is nilmnfd!
Nilmanifolds
Classify M that self-cover with deg > 1.
Problem:
Examples
:
Gromov’s Expanding Maps Theorem [’81]:
f : M ! M expanding self-cover
(up to finite cover)
=) M is nilmnfd!expanding
v
Nilmanifolds
Classify M that self-cover with deg > 1.
Problem:
Examples
:
Gromov’s Expanding Maps Theorem [’81]:
f : M ! M expanding self-cover
v
Df(v)
(up to finite cover)
=) M is nilmnfd!
expanding
Nilmanifolds
Classify M that self-cover with deg > 1.
Problem:
Examples
:
Gromov’s Expanding Maps Theorem [’81]:
f : M ! M expanding self-cover
v
kDf(v)k > kvkDf(v)
(up to finite cover)
=) M is nilmnfd!
expanding
Nilmanifolds
Classify M that self-cover with deg > 1.
Problem:
Examples
:
Gromov’s Expanding Maps Theorem [’81]:
(up to finite cover)
f : M ! M expanding self-cover =) M is nilmnfd!
Nilmanifolds
Classify M that self-cover with deg > 1.
Problem:
Examples
:
Gromov’s Expanding Maps Theorem [’81]:
(up to finite cover)
f : M ! M expanding self-cover =) M is nilmnfd!
Are these all?Q:
Nilmanifolds
Classify M that self-cover with deg > 1.
Problem:
Examples
:
Gromov’s Expanding Maps Theorem [’81]:
(up to finite cover)
f : M ! M expanding self-cover =) M is nilmnfd!
Are these all?Q:
A:T 2 ! M!
S1⌃⇥ S1 ,No! Remember:
Nilmanifolds
Classify M that self-cover with deg > 1.
Problem:
Examples
:
T 2 ! M!S1
⌃⇥ S1 ,
[nilmnfd] ! M#B
Nilmanifolds
1)
2)
3)
Classify M that self-cover with deg > 1.
Problem:
Examples
: [nilmnfd] ! M#B
Ambitious Conj:
Any self-cover is of this form
(up to finite cover).
Classify M that self-cover with deg > 1.
Problem:
Examples
: [nilmnfd] ! M#B
Ambitious Conj:
Any self-cover is of this form
(up to finite cover)
Agol-Teichner-vL: False!
First “exotic” examples.
.
using: Baumslag–Solitar groups,
4–mnfd topology results by Hambleton–Kreck–Teichner[ [
Classify M that self-cover with deg > 1.
Problem:
“coming from symmetry”� G/G
( )regulari.e./ Galois / map is quotient by a group action
New
M
M
Classify M that self-cover with deg > 1.
Problem:
“coming from symmetry”
� G/G
Problem:
Surprisingly mild condition.
( )regular
New
M
M
Classify M that self-cover with deg > 1.
Problem:
“coming from symmetry”
� G/G
Problem:
Surprisingly mild condition.
( )regular
� G/G
M
M
M
Idea: Iterate!
New
Classify M that self-cover with deg > 1.
Problem:
“coming from symmetry”
Problem:
Surprisingly mild condition.
( )regular
Idea: Iterate!
M
M
M
M
M
Define:
all iterates are regular
()strongly regular
New
M
Classify M that self-cover with deg > 1.
Problem:
M
M
Define:
all iterates are regular()strongly regular
Classify strongly reg. self-covers.
New
M
M
M
M
Problem:
M
M
M
M
M
Define:
all iterates are regular()strongly regular
Thm 1 [vL]: On level of ⇡1,
strongly reg. covers come from torus endo’s
Classify strongly reg. self-covers.
:
New
M
Problem:
M
M
M
M
M
Define:
all iterates are regular()strongly regular
Thm 1 [vL]: On level of ⇡1,
strongly reg. covers come from torus endo’s
Classify strongly reg. self-covers.
:
⇡1(M)
#p⇤⇡1(M)
Zk
Zk
⇣
⇣# 9A
9q
New
M
Problem:
Define:
all iterates are regular()strongly regular
Thm 1 [vL]: On level of ⇡1,
strongly reg. covers come from torus endo’s
Classify strongly reg. self-covers.
:
⇡1(M)
#p⇤⇡1(M)
Zk
Zk
⇣
⇣# 9A
9q
New
M
M
M
M
M
M
ker(q)
#ker(q)
!⇠=
!
Define:
all iterates are regular()strongly regular
Thm 1 [vL]: On level of ⇡1,
strongly reg. covers come from torus endo’s
:
⇡1(M)
#p⇤⇡1(M)
Zk
Zk
⇣
⇣# 9A
9q
Thm 2 [vL]: M Kahler,
p : M ! M hol. strongly reg.
=) M ⇠= N ⇥ T (up to finite cover).
M
M
M
M
M
M
ker(q)
#ker(q)
!⇠=
!
Thm 1 [vL]:
⇡1(M) Zk⇣9qp : M ! M strongly reg.
=)Proof idea:
M
M
M
M
M
M
Thm 1 [vL]:
⇡1(M) Zk⇣9qp : M ! M strongly reg.
=)Proof idea:
Step 1: Change perspective.M
M
M
M
M
M
Thm 1 [vL]:
⇡1(M) Zk⇣9qp : M ! M strongly reg.
=)Proof idea:
Step 1: Change perspective.
M
M
M
M
Notation: � := ⇡1(M),
' := p⇤ : � ,! �
��
��
�
�/'(�)
�/'2(�)
�/'3(�)
�/'4(�)
�/'5(�)
M
M
Thm 1 [vL]:
⇡1(M) Zk⇣9qp : M ! M strongly reg.
=)Proof idea:
Step 1: Change perspective.
M
M
M
M
Notation: � := ⇡1(M),
' := p⇤ : � ,! �
��
��
�
�/'(�)
�/'2(�)
�/'3(�)
�/'4(�)
�/'5(�)
M
M
⇥2
M = S1
Thm 1 [vL]:
⇡1(M) Zk⇣9qp : M ! M strongly reg.
=)Proof idea:
Step 1: Change perspective.
M
M
M
M
Notation: � := ⇡1(M),
' := p⇤ : � ,! �
��
��
�
�/'(�)
�/'2(�)
�/'3(�)
�/'4(�)
�/'5(�)
Step 2: Take limit of groups.
M
M
⇥2
M = S1
Thm 1 [vL]:
⇡1(M) Zk⇣9qp : M ! M strongly reg.
=)Proof idea:
Step 1: Change perspective.
M
M
M
M
Notation: � := ⇡1(M),
' := p⇤ : � ,! �
��
��
�
�/'(�)
�/'2(�)
�/'3(�)
�/'4(�)
�/'5(�)
Step 2: Take limit of groups.
,!,!
,!,!
,!
{(direct!)
⇥2
M = S1
⇣:= lim�!�/'n(�)
⌘
(i) Acts on M ,
(ii) Self-similar algebr. struct.
“�/'1”
M
M
Thm 1 [vL]:
⇡1(M) Zk⇣9qp : M ! M strongly reg.
=)Proof idea:
M
M
M
M
��
��
�
�/'(�)
�/'2(�)
�/'3(�)
�/'4(�)
�/'5(�)
,!,!
,!,!
,!
(i) Acts on M ,
(ii) Self-similar algebr. struct.
“�/'1”
⇥2
M = S1
M
M
Thm 1 [vL]:
⇡1(M) Zk⇣9qp : M ! M strongly reg.
=)Proof idea:
M
M
M
M
��
��
�
�/'(�)
�/'2(�)
�/'3(�)
�/'4(�)
�/'5(�)
,!,!
,!,!
,!
Step 3:
Loc. fin. gps + Fin. Gp. Actions
=)F is Artinian
+
(i) Acts on M ,
(ii) Self-similar algebr. struct.
“�/'1”
M
M
Thm 1 [vL]:
⇡1(M) Zk⇣9qp : M ! M strongly reg.
=)Proof idea:
M
M
M
M
��
��
�
�/'(�)
�/'2(�)
�/'3(�)
�/'4(�)
�/'5(�)
,!,!
,!,!
,!
F is Artinian =)ˇ
Sunkov
Kegel-Wehrfritz[ [F is virt. abelian
M
M
Thm 2 [vL]: M Kahler,
p : M ! M hol. strongly reg.
=) M ⇠= N ⇥ T (up to finite cover)
Proof idea:
Thm 2 [vL]: M Kahler,
p : M ! M hol. strongly reg.
=) M ⇠= N ⇥ T (up to finite cover)
Proof idea:
F ✓ Hol(M)
Thm 2 [vL]: M Kahler,
p : M ! M hol. strongly reg.
=) M ⇠= N ⇥ T (up to finite cover)
Proof idea:
F ✓ Hol(M) is a torus T(using Kahler geometry)
Thm 2 [vL]: M Kahler,
p : M ! M hol. strongly reg.
=) M ⇠= N ⇥ T (up to finite cover)
Proof idea:
F ✓ Hol(M) is a torus T(using Kahler geometry)
T y M freelyDi�cult pt:
Thm 2 [vL]: M Kahler,
p : M ! M hol. strongly reg.
=) M ⇠= N ⇥ T (up to finite cover)
Proof idea:
Lift to
eT y fM�conj
by ep
F ✓ Hol(M) is a torus T(using Kahler geometry)
T y M freelyDi�cult pt:
Geom. linear map
Thm 2 [vL]: M Kahler,
p : M ! M hol. strongly reg.
=) M ⇠= N ⇥ T (up to finite cover)
Proof idea:
Lift to
eT y fM�conj
by ep
F ✓ Hol(M) is a torus T(using Kahler geometry)
T y M freelyDi�cult pt:
Thm 1 ⇡1(M)
#p⇤⇡1(M)
Zk
Zk
⇣
⇣# 9A
9qGeom. linear map
Thm 2 [vL]: M Kahler,
p : M ! M hol. strongly reg.
=) M ⇠= N ⇥ T (up to finite cover)
Proof idea:
Lift to
eT y fM�conj
by epGeom. linear map
F ✓ Hol(M) is a torus T(using Kahler geometry)
T y M freelyDi�cult pt:
Thm 1 Zk
Zk
# 9AAlg. linear map
Thm 2 [vL]: M Kahler,
p : M ! M hol. strongly reg.
=) M ⇠= N ⇥ T (up to finite cover)
Proof idea:
Lift to
eT y fM�conj
by epGeom. linear map
F ✓ Hol(M) is a torus T(using Kahler geometry)
T y M freelyDi�cult pt:
Thm 1 Zk
Zk
# 9AAlg. linear map=
KEY!
Problem:
Which Riemannian manifolds X have
9µ > 0 : 8� : vol(X/�) � µ
“minimal quotients”:
?
Earlier: R2 NO H2 YES
#
(Image: Claudio Rocchini)
,
Problem:
Which Riemannian manifolds X have
9µ > 0 : 8� : vol(X/�) � µ
“minimal quotients”:
?
Earlier:
Thm (Kazhdan-Margulis, 1968):
G semisimple (e.g. SL(n,R))
=) G and G/K have min’l quot’s
R2 NO H2 YES,
Problem:
Which Riemannian manifolds X have
9µ > 0 : 8� : vol(X/�) � µ ?Thm (Kazhdan-Margulis, 1968):
G semisimple (e.g. SL(n,R))
=) G and G/K have min’l quot’s
Conj (Margulis, 1974 ICM):
X: �1 K 0, no Eucl factors=) min’l quot’s
(vol � µ(dimX))
Problem:
Which Riemannian manifolds X have
9µ > 0 : 8� : vol(X/�) � µ ?Thm (Kazhdan-Margulis, 1968):
G semisimple (e.g. SL(n,R))
=) G and G/K have min’l quot’s
Conj (Margulis, 1974 ICM):
X: �1 K 0, no Eucl factors=) min’l quot’s
(vol � µ(dimX))
Gromov [’78]: �1 K < 0
(e.g. SL(n,R)) =) G and G/K have min’l quot’s
Conj (Margulis, 1974 ICM):
X: �1 K 0, no Eucl factors=) min’l quot’s
(vol � µ(dimX))
Gromov [’78]: �1 K < 0
X contractible with some cmpt quot M
⇡1(M) no normal abelian subgps
Then 8 metric, �: vol(X/�) � µdimRic
injraddiam[ [
G semisimple
Thm (Kazhdan-Margulis, 1968)
Thm 3 [vL]
(e.g. SL(n,R)) =) G and G/K have min’l quot’s
Conj (Margulis, 1974 ICM):
X: �1 K 0, no Eucl factors=) min’l quot’s
(vol � µ(dimX))
Gromov [’78]: �1 K < 0
X contractible with some cmpt quot M
⇡1(M) no normal abelian subgps
Then 8 metric, �: vol(X/�) � µdimRic
injraddiam[ [
G semisimple
Thm 3 [vL]
Thm (Kazhdan-Margulis, 1968)
Thm 3 [vL]
Thm (Kazhdan-Margulis, 1968)
w/o Eucl. factors
}Topo-
logy!
e.g. K 0
(e.g. SL(n,R)) =) G and G/K have min’l quot’s
Conj (Margulis, 1974 ICM):
X: �1 K 0, no Eucl factors=) min’l quot’s
(vol � µ(dimX))
Gromov [’78]: �1 K < 0
X contractible with some cmpt quot M
⇡1(M) no normal abelian subgps
Then 8 metric, �: vol(X/�) � µdimRic
injraddiam[ [
G semisimple
Remark:
=)
Thm 3 [vL]
Thm (Kazhdan-Margulis, 1968)
Thm 3 [vL] Thm (Kazhdan-Margulis, 1968)
w/o Eucl. factors
}Topo-
logy!
e.g. K 0
X contractible with some cmpt quot M
⇡1(M) no normal abelian subgps
Then 8 metric, �: vol(X/�) � µdimRic
injraddiam[ [
Thm 3 [vL]
Proof idea:
X contractible with some cmpt quot M
⇡1(M) no normal abelian subgps
Then 8 metric, �: vol(X/�) � µdimRic
injraddiam[ [
Proof idea:
Suppose vol(X/�n) ! 0.
Geom. bounds =) gn ! g (Cheeger–Anderson)
Thm 3 [vL]
X contractible with some cmpt quot M
⇡1(M) no normal abelian subgps
Then 8 metric, �: vol(X/�) � µdimRic
injraddiam[ [
Proof idea:
Suppose vol(X/�n) ! 0.
Geom. bounds =) gn ! g (Cheeger–Anderson)
Hard: �n �! G(discrete) (cnts)
Thm 3 [vL]
Z2
Z212
1nZ
2 ! R2
(discrete) (cnts)
X contractible with some cmpt quot M
⇡1(M) no normal abelian subgps
Then 8 metric, �: vol(X/�) � µdimRic
injraddiam[ [
Proof idea:
Suppose vol(X/�n) ! 0.
Geom. bounds =) gn ! g (Cheeger–Anderson)
Hard: �n �! G(discrete) (cnts)
Show: G is semisimple Lie.
Thm 3 [vL]
Common framework:
M � Isom(M)
“symmetry”
Common framework:
M � Isom(M)
“symmetry”
Isom(M1) Isom(M2)
Isom(
fM)
fM
M1 M2 �
�
�
Common framework:
M � Isom(M)
“symmetry”
“Hidden Symmetries”
Isom(M1) Isom(M2)
Isom(
fM)
fM
M1 M2 �
�
�
Lie group with natural lattice
acting on a manifold
⇡1(M) ✓ Isom(
fM) � fM
