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