Symmetry and self- similarity in geometry Wouter van Limbeek - - PowerPoint PPT Presentation

Symmetry and self- similarity in geometry

Wouter van Limbeek University of Michigan University of Cambridge 29 January 2018

T 2 =

1 1

An example

T 2 =

1 1

1/n 1/n

An example

T 2 =

1 1

1/n 1/n

An example

T 2 =

1 1

1/n 1/n

An example

An example

T 2 = T 2 =

1 1

1/n 1/n

n2 : 1

An example

T 2 T 2

1 1

1/n 1/n

n2 : 1

• 1. ∃ covers

with degree > 1

• 2. Symmetry

covers Remarks:

Genus 2

≥

∀g ≥ 2 : @ covers Σg → Σg χ = 2 2g 6= 0 Hurwitz’s 84(g − 1) Theorem (1893): Reason: 1. 2. Σg Riemann surface, g ≥ 2 = ⇒ |Aut+(Σg)| ≤ 84(g − 1)

Genus 2

≥

Hurwitz’s 84(g − 1) Theorem (1893): |Aut+(Σg)| ≤ 84(g − 1)

Genus 2

≥

Hurwitz’s 84(g − 1) Theorem (1893): |Aut+(Σg)| ≤ 84(g − 1)

∀X = H2/Γ : Area(X) ≥ π

21

Key fact [

area estimate for “orbifold” quotients

:

[

↓

(Image: Claudio Rocchini)

Genus 2

≥

Hurwitz’s 84(g − 1) Theorem (1893): |Aut+(Σg)| ≤ 84(g − 1)

∀X = H2/Γ : Area(X) ≥ π

21

Key fact [

area estimate for “orbifold” quotients

:

[

(Image: Claudio Rocchini)

Area(Σg/Aut+(Σg)) =

Area(Σg) |Aut+(Σg)|

≥ π/21

Connection:

↓

Two basic problems

∃µ > 0 : ∀Γ : vol(X/Γ) ≥ µ ?

1.

↓

1 1

1/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] : Σ × 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] : Σ × S1, T 2 → M → S1

K¨ ahler and dimC = 2, 3 [H¨

• ring-Peternell, ’11]

Classify M that self-cover with deg > 1.

Problem:

dim = 2 : T 2, K

Low dimensions:

dim = 3 [Yu-Wang ’99] : Σ × S1, T 2 → M → S1

K¨ ahler and dimC = 2, 3 [H¨

• ring-Peternell, ’11]

dim ≥ 4

• 1. Tori T n = Rn/Zn
• A ∈ 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] : Σ × S1, T 2 → M → S1

K¨ ahler and dimC = 2, 3 [H¨

• ring-Peternell, ’11]

dim ≥ 4

• 1. Tori T n = Rn/Zn
• A ∈ 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 > kvk 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]:

(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 Σ × S1

, No! Remember: Nilmanifolds

Classify M that self-cover with deg > 1.

Problem:

Examples:

T 2 → M → S1 Σ × 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

( ) regular i.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 π1, 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 π1, strongly reg. covers come from torus endo’s Classify strongly reg. self-covers. : π1(M)

↓p∗

π1(M) Zk Zk

⇣ ⇣

↓ ∃A

∃q

New

M

Problem:

Define: all iterates are regular ⇐ ⇒ strongly regular Thm 1 [vL]: On level of π1, strongly reg. covers come from torus endo’s Classify strongly reg. self-covers. : π1(M)

↓p∗

π1(M) Zk Zk

⇣ ⇣

↓ ∃A

∃q

New

M M M M M M

ker(q)

↓

ker(q)

→

∼ =

→

Define: all iterates are regular ⇐ ⇒ strongly regular Thm 1 [vL]: On level of π1, strongly reg. covers come from torus endo’s: π1(M)

↓p∗

π1(M) Zk Zk

⇣ ⇣

↓ ∃A

∃q Thm 2 [vL]: M K¨ ahler, 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]: π1(M) Zk

⇣

∃q p : M → M strongly reg.

= ⇒

Proof idea:

M M M M M M

Thm 1 [vL]: π1(M) Zk

⇣

∃q p : M → M strongly reg.

= ⇒

Proof idea: Step 1: Change perspective.

M M M M M M

Thm 1 [vL]: π1(M) Zk

⇣

∃q p : M → M strongly reg.

= ⇒

Proof idea: Step 1: Change perspective.

M M M M

Notation: Γ := π1(M), ' := p∗ : Γ , → Γ

• Γ/ϕ(Γ)

Γ/ϕ2(Γ) Γ/ϕ3(Γ) Γ/ϕ4(Γ) Γ/ϕ5(Γ)

M M

Thm 1 [vL]: π1(M) Zk

⇣

∃q p : M → M strongly reg.

= ⇒

Proof idea: Step 1: Change perspective.

M M M M

Notation: Γ := π1(M), ' := p∗ : Γ , → Γ

• Γ/ϕ(Γ)

Γ/ϕ2(Γ) Γ/ϕ3(Γ) Γ/ϕ4(Γ) Γ/ϕ5(Γ)

M M

×2

M = S1

Thm 1 [vL]: π1(M) Zk

⇣

∃q p : M → M strongly reg.

= ⇒

Proof idea: Step 1: Change perspective.

M M M M

Notation: Γ := π1(M), ' := p∗ : Γ , → Γ

• Γ/ϕ(Γ)

Γ/ϕ2(Γ) Γ/ϕ3(Γ) Γ/ϕ4(Γ) Γ/ϕ5(Γ)

Step 2: Take limit of groups.

M M

×2

M = S1

Thm 1 [vL]: π1(M) Zk

⇣

∃q p : M → M strongly reg.

= ⇒

Proof idea: Step 1: Change perspective.

M M M M

Notation: Γ := π1(M), ' := p∗ : Γ , → Γ

• Γ/ϕ(Γ)

Γ/ϕ2(Γ) Γ/ϕ3(Γ) Γ/ϕ4(Γ) Γ/ϕ5(Γ)

Step 2: Take limit of groups.

, → , → , → , → , →

{

(direct!)

×2

M = S1 ⇣ := lim − → Γ/ϕn(Γ) ⌘

(i) Acts on M, (ii) Self-similar algebr. struct. “Γ/ϕ∞”

M M

Thm 1 [vL]: π1(M) Zk

⇣

∃q p : M → M strongly reg.

= ⇒

Proof idea:

M M M M

• Γ/ϕ(Γ)

Γ/ϕ2(Γ) Γ/ϕ3(Γ) Γ/ϕ4(Γ) Γ/ϕ5(Γ)

, → , → , → , → , →

(i) Acts on M, (ii) Self-similar algebr. struct. “Γ/ϕ∞”

×2

M = S1

M M

Thm 1 [vL]: π1(M) Zk

⇣

∃q p : M → M strongly reg.

= ⇒

Proof idea:

M M M M

• Γ/ϕ(Γ)

Γ/ϕ2(Γ) Γ/ϕ3(Γ) Γ/ϕ4(Γ) Γ/ϕ5(Γ)

, → , → , → , → , →

Step 3:

• Loc. fin. gps

+ Fin. Gp. Actions

= ⇒

F is Artinian + (i) Acts on M, (ii) Self-similar algebr. struct. “Γ/ϕ∞”

M M

Thm 1 [vL]: π1(M) Zk

⇣

∃q p : M → M strongly reg.

= ⇒

Proof idea:

M M M M

• Γ/ϕ(Γ)

Γ/ϕ2(Γ) Γ/ϕ3(Γ) Γ/ϕ4(Γ) Γ/ϕ5(Γ)

, → , → , → , → , →

F is Artinian =

⇒

ˇ Sunkov

Kegel-Wehrfritz

[ [

F is virt. abelian

M M

Thm 2 [vL]: M K¨ ahler, p : M → M hol. strongly reg. = ⇒ M ∼ = N × T

(up to finite cover)

Proof idea:

Thm 2 [vL]: M K¨ ahler, p : M → M hol. strongly reg. = ⇒ M ∼ = N × T

(up to finite cover)

Proof idea: F ⊆ Hol(M)

Thm 2 [vL]: M K¨ ahler, p : M → M hol. strongly reg. = ⇒ M ∼ = N × T

(up to finite cover)

Proof idea: F ⊆ Hol(M) is a torus T

(using K¨ ahler geometry)

Thm 2 [vL]: M K¨ ahler, p : M → M hol. strongly reg. = ⇒ M ∼ = N × T

(up to finite cover)

Proof idea: F ⊆ Hol(M) is a torus T

(using K¨ ahler geometry)

T y M freely Difficult pt:

Thm 2 [vL]: M K¨ ahler, p : M → M hol. strongly reg. = ⇒ M ∼ = N × T

(up to finite cover)

Proof idea: Lift to e T y f M

• conj

by e p

F ⊆ Hol(M) is a torus T

(using K¨ ahler geometry)

T y M freely Difficult pt:

• Geom. linear map

Thm 2 [vL]: M K¨ ahler, p : M → M hol. strongly reg. = ⇒ M ∼ = N × T

(up to finite cover)

Proof idea: Lift to e T y f M

• conj

by e p

F ⊆ Hol(M) is a torus T

(using K¨ ahler geometry)

T y M freely Difficult pt: Thm 1

π1(M)

↓p∗

π1(M) Zk Zk

⇣ ⇣

↓ ∃A

∃q

• Geom. linear map

Thm 2 [vL]: M K¨ ahler, p : M → M hol. strongly reg. = ⇒ M ∼ = N × T

(up to finite cover)

Proof idea: Lift to e T y f M

• conj

by e p

• Geom. linear map

F ⊆ Hol(M) is a torus T

(using K¨ ahler geometry)

T y M freely Difficult pt: Thm 1

Zk Zk

↓ ∃A

• Alg. linear map

Thm 2 [vL]: M K¨ ahler, p : M → M hol. strongly reg. = ⇒ M ∼ = N × T

(up to finite cover)

Proof idea: Lift to e T y f M

• conj

by e p

• Geom. linear map

F ⊆ Hol(M) is a torus T

(using K¨ ahler geometry)

T y M freely Difficult pt: Thm 1

Zk Zk

↓ ∃A

• Alg. linear map

=

KEY!

Problem: Which Riemannian manifolds X have

∃µ > 0 : ∀Γ : vol(X/Γ) ≥ µ

“minimal quotients”:

?

Earlier: R2 NO H2 YES

↓

(Image: Claudio Rocchini)

,

Problem: Which Riemannian manifolds X have

∃µ > 0 : ∀Γ : vol(X/Γ) ≥ 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

∃µ > 0 : ∀Γ : vol(X/Γ) ≥ 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(dim X))

Problem: Which Riemannian manifolds X have

∃µ > 0 : ∀Γ : vol(X/Γ) ≥ 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(dim X)) 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(dim X)) Gromov [’78]: −1 ≤ K < 0 X contractible with some cmpt quot M π1(M) no normal abelian subgps Then ∀ metric, Γ: vol(X/Γ) ≥ µ

dim Ric injrad diam

[ [

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(dim X)) Gromov [’78]: −1 ≤ K < 0 X contractible with some cmpt quot M π1(M) no normal abelian subgps Then ∀ metric, Γ: vol(X/Γ) ≥ µ

dim Ric injrad diam

[ [

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(dim X)) Gromov [’78]: −1 ≤ K < 0 X contractible with some cmpt quot M π1(M) no normal abelian subgps Then ∀ metric, Γ: vol(X/Γ) ≥ µ

dim Ric injrad diam

[ [

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 π1(M) no normal abelian subgps Then ∀ metric, Γ: vol(X/Γ) ≥ µ

dim Ric injrad diam

[ [

Thm 3 [vL] Proof idea:

X contractible with some cmpt quot M π1(M) no normal abelian subgps Then ∀ metric, Γ: vol(X/Γ) ≥ µ

dim Ric injrad diam

[ [

Proof idea: Suppose vol(X/Γn) → 0.

• Geom. bounds =

⇒ gn → g (Cheeger–Anderson) Thm 3 [vL]

X contractible with some cmpt quot M π1(M) no normal abelian subgps Then ∀ metric, Γ: vol(X/Γ) ≥ µ

dim Ric injrad diam

[ [

Proof idea: Suppose vol(X/Γn) → 0.

• Geom. bounds =

⇒ gn → g (Cheeger–Anderson) Hard: Γn − → G

(discrete) (cnts)

Thm 3 [vL]

Z2

Z2

1 2

1 nZ2 → R2

(discrete) (cnts)

X contractible with some cmpt quot M π1(M) no normal abelian subgps Then ∀ metric, Γ: vol(X/Γ) ≥ µ

dim Ric injrad diam

[ [

Proof idea: Suppose vol(X/Γn) → 0.

• Geom. bounds =

⇒ gn → g (Cheeger–Anderson) Hard: Γn − → 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(f M) f M M1 M2

Common framework:

M

• Isom(M)

“symmetry” “Hidden Symmetries” Isom(M1) Isom(M2) Isom(f M) f M M1 M2

• Lie group with natural lattice

acting on a manifold π1(M) ⊆ Isom(f M) f M