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

An example 1 T 2 = 0 1

An example 1 T 2 = 1 /n 1 /n 0 1

An example 1 T 2 = 1 /n 1 /n 0 1

An example 1 T 2 = 1 /n 1 /n 0 1

An example 1 T 2 = 1 /n 1 /n 0 1 n 2 : 1 T 2 =

An example 1 Remarks: T 2 1. ∃ covers with degree > 1 1 /n 1 /n T 2 0 1 n 2 : 1 2. Symmetry covers

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

Genus 2 ≥ Hurwitz’s 84( g − 1) Theorem (1893): | Aut + ( Σ g ) | ≤ 84( g − 1)

Genus 2 ≥ Hurwitz’s 84( g − 1) Theorem (1893): | Aut + ( Σ g ) | ≤ 84( g − 1) ↓ Key fact [ area estimate for [ : “orbifold” quotients ∀ X = H 2 / Γ : Area( X ) ≥ π 21 (Image: Claudio Rocchini)

Genus 2 ≥ Hurwitz’s 84( g − 1) Theorem (1893): | Aut + ( Σ g ) | ≤ 84( g − 1) ↓ Key fact [ area estimate for [ : “orbifold” quotients ∀ X = H 2 / Γ : Area( X ) ≥ π 21 Connection: Area( Σ g ) Area( Σ g / Aut + ( Σ g )) = | Aut + ( Σ g ) | ≥ π / 21 (Image: Claudio Rocchini)

Two basic problems 1 1 /n ↓ 1 /n 0 1 n 2 : 1 Classify M that Which Riem. mnfds X 1. 2. self-cover with deg > 1. have “minimal quotients”: ∃ µ > 0 : ∀ Γ : vol( X/ Γ ) ≥ µ ?

Problem: Classify M that self-cover with deg > 1.

Problem: Classify M that self-cover with deg > 1. Low dimensions: dim = 2 : T 2 , K

Problem: Classify M that self-cover with deg > 1. Low dimensions: dim = 2 : T 2 , K T 2 → M dim = 3 [Yu-Wang ’99] : Σ × S 1 , → S 1

Problem: Classify M that self-cover with deg > 1. Low dimensions: dim = 2 : T 2 , K T 2 → M dim = 3 [Yu-Wang ’99] : Σ × S 1 , → S 1 K¨ ahler and dim C = 2 , 3 [H¨ oring-Peternell, ’11]

Problem: Classify M that self-cover with deg > 1. Low dimensions: dim = 2 : T 2 , K T 2 → M dim = 3 [Yu-Wang ’99] : Σ × S 1 , → S 1 K¨ ahler and dim C = 2 , 3 [H¨ oring-Peternell, ’11] ( ) dim ≥ 4 Examples 1. Tori T n = R n / Z n � A ∈ M n ( Z ) , deg = | det( A ) |

Problem: Classify M that self-cover with deg > 1. Low dimensions: dim = 2 : T 2 , K T 2 → M dim = 3 [Yu-Wang ’99] : Σ × S 1 , → S 1 K¨ ahler and dim C = 2 , 3 [H¨ oring-Peternell, ’11] ( ) dim ≥ 4 Examples 1. Tori T n = R n / Z n � A ∈ M n ( Z ) , deg = | det( A ) | 2. Nilmanifolds

Problem: Classify M that self-cover with deg > 1. Examples : Nilmanifolds

Problem: Classify M that self-cover with deg > 1. Examples : Nilmanifolds Gromov’s Expanding Maps Theorem [’81] : f : M → M expanding self-cover = ⇒ M is nilmnfd! (up to finite cover)

Problem: Classify M that self-cover with deg > 1. Examples : Nilmanifolds Gromov’s Expanding Maps Theorem [’81] : f : M → M expanding self-cover expanding = ⇒ M is nilmnfd! (up to finite cover) v

Problem: Classify M that self-cover with deg > 1. Examples : Nilmanifolds Gromov’s Expanding Maps Theorem [’81] : f : M → M expanding self-cover = ⇒ M is nilmnfd! (up to finite cover) v expanding Df ( v )

Problem: Classify M that self-cover with deg > 1. Examples : Nilmanifolds Gromov’s Expanding Maps Theorem [’81] : f : M → M expanding self-cover = ⇒ M is nilmnfd! (up to finite cover) v expanding Df ( v ) k Df ( v ) k > k v k

Problem: Classify M that self-cover with deg > 1. Examples : Nilmanifolds Gromov’s Expanding Maps Theorem [’81] : f : M → M expanding self-cover = ⇒ M is nilmnfd! (up to finite cover)

Problem: Classify M that self-cover with deg > 1. Examples : Nilmanifolds Gromov’s Expanding Maps Theorem [’81] : f : M → M expanding self-cover = ⇒ M is nilmnfd! (up to finite cover) Q: Are these all?

Problem: Classify M that self-cover with deg > 1. Examples : Nilmanifolds Gromov’s Expanding Maps Theorem [’81] : f : M → M expanding self-cover = ⇒ M is nilmnfd! (up to finite cover) Q: Are these all? T 2 → M → A: No! Remember: Σ × S 1 , S 1

Problem: Classify M that self-cover with deg > 1. 1) Examples : Nilmanifolds T 2 → M → 2) Σ × S 1 , S 1 [nilmnfd] → M ↓ 3) B

Problem: Classify M that self-cover with deg > 1. Examples : [nilmnfd] → M ↓ B Ambitious Conj: Any self-cover is of this form (up to finite cover) .

Problem: Classify M that self-cover with deg > 1. 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

New Problem: Classify M that self-cover with deg > 1. “coming from symmetry” M G � ( / Galois / map is quotient by a group action ) i.e. regular /G M

New Problem: Classify M that self-cover with deg > 1. “coming from symmetry” ( ) regular M G � Problem: Surprisingly mild condition. /G M

New Problem: Classify M that self-cover with deg > 1. “coming from symmetry” ( ) regular M G � Problem: Surprisingly mild condition. Idea: Iterate! /G M G � /G M

New Problem: Classify M that self-cover with deg > 1. “coming from symmetry” M ( ) regular Problem: Surprisingly mild condition. M Idea: Iterate! M Define: strongly regular M ⇐ ⇒ all iterates are regular M M

Define: strongly regular all iterates are regular ⇐ ⇒ M Problem: New M Classify strongly reg. self-covers. M M M M

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

Define: strongly regular all iterates are regular ⇐ ⇒ M Problem: New M Classify strongly reg. self-covers. Thm 1 [vL]: On level of π 1 , M strongly reg. covers come from torus endo’s : M ∃ q Z k π 1 ( M ) ⇣ ↓ ∃ A ↓ p ∗ M Z k π 1 ( M ) ⇣ M

Define: strongly regular all iterates are regular ⇐ ⇒ M Problem: New M Classify strongly reg. self-covers. Thm 1 [vL]: On level of π 1 , M strongly reg. covers come from torus endo’s : M ∃ q Z k ker( q ) π 1 ( M ) ⇣ → ↓ ∃ A ↓ ↓ p ∗ ∼ = M Z k ker( q ) π 1 ( M ) ⇣ → M

Define: strongly regular all iterates are regular ⇐ ⇒ M Thm 1 [vL]: On level of π 1 , strongly reg. covers come from torus endo’s : M ∃ q Z k π 1 ( M ) ker( q ) ⇣ → M ↓ ∃ A ↓ p ∗ ↓ ∼ = Z k π 1 ( M ) ker( q ) ⇣ → M Thm 2 [vL]: M K¨ ahler, M p : M → M hol. strongly reg. ⇒ M ∼ = N × T (up to finite cover) . M =

Thm 1 [vL]: p : M → M strongly reg. ∃ q Z k ⇒ π 1 ( M ) = ⇣ M Proof idea: M M M M M

Thm 1 [vL]: p : M → M strongly reg. ∃ q Z k ⇒ π 1 ( M ) = ⇣ M Proof idea: M Step 1: Change perspective. M M M M

Thm 1 [vL]: p : M → M strongly reg. � M Γ / ϕ 5 ( Γ ) ∃ q Z k ⇒ π 1 ( M ) = ⇣ � Γ / ϕ 4 ( Γ ) M Proof idea: � Step 1: Change perspective. Γ / ϕ 3 ( Γ ) M Notation: Γ := π 1 ( M ), ' := p ∗ : Γ , → Γ � Γ / ϕ 2 ( Γ ) M � Γ / ϕ ( Γ ) M M

Thm 1 [vL]: p : M → M strongly reg. � M Γ / ϕ 5 ( Γ ) ∃ q Z k ⇒ π 1 ( M ) = ⇣ � Γ / ϕ 4 ( Γ ) M Proof idea: � Step 1: Change perspective. Γ / ϕ 3 ( Γ ) M Notation: Γ := π 1 ( M ), ' := p ∗ : Γ , → Γ � Γ / ϕ 2 ( Γ ) M � Γ / ϕ ( Γ ) M M M = S 1 × 2

Thm 1 [vL]: p : M → M strongly reg. � M Γ / ϕ 5 ( Γ ) ∃ q Z k ⇒ π 1 ( M ) = ⇣ � Γ / ϕ 4 ( Γ ) M Proof idea: � Step 1: Change perspective. Γ / ϕ 3 ( Γ ) M Notation: Γ := π 1 ( M ), ' := p ∗ : Γ , → Γ � Γ / ϕ 2 ( Γ ) M Step 2: Take limit of groups. � Γ / ϕ ( Γ ) M M M = S 1 × 2

→ , Thm 1 [vL]: p : M → M strongly reg. � M Γ / ϕ 5 ( Γ ) → ∃ q Z k ⇒ π 1 ( M ) , = ⇣ � Γ / ϕ 4 ( Γ ) M → Proof idea: , � Step 1: Change perspective. Γ / ϕ 3 ( Γ ) M → Notation: Γ := π 1 ( M ), , ' := p ∗ : Γ , → Γ � Γ / ϕ 2 ( Γ ) M → Step 2: Take limit of groups. , { � Γ / ϕ ( Γ ) M (direct!) ⇣ ⌘ “ Γ / ϕ ∞ ” := lim → Γ / ϕ n ( Γ ) − M (i) Acts on M , M = S 1 (ii) Self-similar algebr. struct. × 2

→ , Thm 1 [vL]: p : M → M strongly reg. � M Γ / ϕ 5 ( Γ ) → ∃ q Z k ⇒ π 1 ( M ) , = ⇣ � Γ / ϕ 4 ( Γ ) M → Proof idea: “ Γ / ϕ ∞ ” , � Γ / ϕ 3 ( Γ ) M (i) Acts on M , → (ii) Self-similar algebr. struct. , � Γ / ϕ 2 ( Γ ) M → , � Γ / ϕ ( Γ ) M M M = S 1 × 2

→ , Thm 1 [vL]: p : M → M strongly reg. � M Γ / ϕ 5 ( Γ ) → ∃ q Z k ⇒ π 1 ( M ) , = ⇣ � Γ / ϕ 4 ( Γ ) M → Proof idea: “ Γ / ϕ ∞ ” , � Γ / ϕ 3 ( Γ ) M (i) Acts on M , → (ii) Self-similar algebr. struct. , � Γ / ϕ 2 ( Γ ) M + Step 3: → , Loc. fin. gps + Fin. Gp. Actions � Γ / ϕ ( Γ ) M ⇒ = M F is Artinian