Symmetry gaps in Riemannian geometry and minimal orbifolds Symmetry gaps in Riemannian geometry Wouter van Limbeek and minimal orbifolds Wouter van Limbeek University of Chicago limbeek @ math.uchicago.edu March 15, 2015
A geometric dichotomy Symmetry gaps in Riemannian geometry and minimal orbifolds Wouter van Limbeek
A geometric dichotomy Symmetry gaps in Riemannian geometry and minimal orbifolds Wouter van Limbeek
A geometric dichotomy Symmetry gaps in Riemannian geometry and minimal orbifolds Wouter van Limbeek
A geometric dichotomy Symmetry gaps in Riemannian geometry and minimal orbifolds Wouter van Limbeek Question Given ( M , g ), can we bound | Isom ( M , g ) | ?
History Symmetry Author Manifold Bound gaps in Riemannian geometry and minimal orbifolds Wouter van Limbeek
History Symmetry Author Manifold Bound gaps in Riemannian Hurwitz Σ g 84( g − 1) geometry and minimal orbifolds Wouter van Limbeek
History Symmetry Author Manifold Bound gaps in Riemannian Hurwitz Σ g 84( g − 1) geometry and minimal orbifolds Bochner-Yano Ric < 0 < ∞ Wouter van Limbeek
History Symmetry Author Manifold Bound gaps in Riemannian Hurwitz Σ g 84( g − 1) geometry and minimal orbifolds Bochner-Yano Ric < 0 < ∞ Wouter van Limbeek Kazhdan-Margulis Locally symmetric C ( n ) vol ( M ) space (e.g. hyperbolic)
History Symmetry Author Manifold Bound gaps in Riemannian Hurwitz Σ g 84( g − 1) geometry and minimal orbifolds Bochner-Yano Ric < 0 < ∞ Wouter van Limbeek Kazhdan-Margulis Locally symmetric C ( n ) vol ( M ) space (e.g. hyperbolic) Gromov K < 0 Dimension κ where K ≤ − κ 2 < 0 Volume
History Symmetry Author Manifold Bound gaps in Riemannian Hurwitz Σ g 84( g − 1) geometry and minimal orbifolds Bochner-Yano Ric < 0 < ∞ Wouter van Limbeek Kazhdan-Margulis Locally symmetric C ( n ) vol ( M ) space (e.g. hyperbolic) Gromov K < 0 Dimension κ where K ≤ − κ 2 < 0 Volume Dai-Shen-Wei Ric < 0 Dimension Ric Injectivity radius Diameter
First theorem Question Symmetry gaps in What if M is not Ricci negatively curved? Riemannian geometry and minimal orbifolds Wouter van Limbeek
First theorem Question Symmetry gaps in What if M is not Ricci negatively curved? Riemannian geometry and minimal An obstruction: orbifolds S 1 � M � No bound on | Isom ( M , g ) | Wouter van Limbeek
First theorem Question Symmetry gaps in What if M is not Ricci negatively curved? Riemannian geometry and minimal An obstruction: orbifolds S 1 � M � No bound on | Isom ( M , g ) | Wouter van Limbeek Theorem (vL, 2014) Let M n be a closed Riemannian manifold, such that | Ric ( M ) | ≤ Λ , injrad ( M ) ≥ ε , diam ( M ) ≤ D, M does not admit an S 1 -action. Then | Isom ( M ) | ≤ C ( n , Λ , ε, D ) .
More general problem Symmetry Lift to the universal cover: gaps in Riemannian geometry and minimal orbifolds Wouter van Limbeek
More general problem Symmetry Lift to the universal cover: gaps in Riemannian geometry and minimal orbifolds Wouter van Limbeek Question Given ( M , g ), can we bound | Isom ( M , g ) | ?
More general problem Symmetry Lift to the universal cover: gaps in Riemannian geometry and minimal orbifolds Wouter van Limbeek Question Given ( M , g ), can we bound [ Isom ( � M , � g ) : π 1 ( M )]?
Higher genus surface Symmetry Let M = Σ g , g ≥ 2. gaps in Riemannian geometry Theorem (Hurwitz) and minimal orbifolds | Isom (Σ g ) | ≤ 84( g − 1) . Wouter van Limbeek
Higher genus surface Symmetry Let M = Σ g , g ≥ 2. gaps in Riemannian geometry Theorem (Hurwitz) and minimal orbifolds | Isom (Σ g ) | ≤ 84( g − 1) . Wouter van Limbeek However: Example [ Isom ( H 2 ) : π 1 (Σ g )] = ∞ .
Higher genus surface Symmetry Let M = Σ g , g ≥ 2. gaps in Riemannian geometry Theorem (Hurwitz) and minimal orbifolds | Isom (Σ g ) | ≤ 84( g − 1) . Wouter van Limbeek However: Example [ Isom ( H 2 ) : π 1 (Σ g )] = ∞ . = ⇒ Ric ( M ) < 0 does not yield a bound!
Second theorem Symmetry gaps in Theorem (vL, 2014) Riemannian geometry Let M n be a closed Riemannian manifold, such that and minimal orbifolds Wouter van Limbeek | Ric ( M ) | ≤ Λ , injrad ( M ) ≥ ε , diam ( M ) ≤ D. � M does not admit a proper action by a nondiscrete Lie group G such that π 1 ( M ) ⊆ G. Then | Isom ( M ) | ≤ C ( n , Λ , ε, D ) .
⇒ locally symmetric Local symmetry = Symmetry Theorem (Farb-Weinberger, 2008) gaps in Riemannian Let M be geometry and minimal orbifolds a closed, aspherical manifold, and not virtually a product, Wouter van π 1 ( M ) has no nontrivial normal abelian subgroups. Limbeek
⇒ locally symmetric Local symmetry = Symmetry Theorem (Farb-Weinberger, 2008) gaps in Riemannian Let M be geometry and minimal orbifolds a closed, aspherical manifold, and not virtually a product, Wouter van π 1 ( M ) has no nontrivial normal abelian subgroups. Limbeek Then TFAE [ Isom ( � M , � g ) , π 1 ( M )] = ∞ , ( M , g ) is isometric to a locally symmetric space.
⇒ locally symmetric Local symmetry = Symmetry Theorem (Farb-Weinberger, 2008) gaps in Riemannian Let M be geometry and minimal orbifolds a closed, aspherical manifold, and not virtually a product, Wouter van π 1 ( M ) has no nontrivial normal abelian subgroups. Limbeek Then TFAE [ Isom ( � M , � g ) , π 1 ( M )] = ∞ , ( M , g ) is isometric to a locally symmetric space. Conjecture (Farb-Weinberger, 2008) For the conclusion above, it suffices that [ Isom ( � M , � g ) : π 1 ( M )] ≥ C for some C only depending on M .
⇒ locally symmetric Local symmetry = Symmetry Theorem (Farb-Weinberger, 2008) gaps in Riemannian Let M be geometry and minimal orbifolds a closed, aspherical manifold, and not virtually a product, Wouter van π 1 ( M ) has no nontrivial normal abelian subgroups. Limbeek Then TFAE [ Isom ( � M , � g ) , π 1 ( M )] = ∞ , ( M , g ) is isometric to a locally symmetric space. Conjecture (Farb-Weinberger, 2008) For the conclusion above, it suffices that [ Isom ( � M , � g ) : π 1 ( M )] ≥ C for some C only depending on M . Theorem (Farb-Weinberger, 2008) True if M is diffeomorphic to a locally symmetric space.
⇒ locally symmetric Local symmetry = Symmetry gaps in Theorem (vL, 2014) Riemannian geometry There exists C ( n , Λ , ε, D ) such that if M n is as in the and minimal orbifolds conjecture, and Wouter van Limbeek
⇒ locally symmetric Local symmetry = Symmetry gaps in Theorem (vL, 2014) Riemannian geometry There exists C ( n , Λ , ε, D ) such that if M n is as in the and minimal orbifolds conjecture, and Wouter van Limbeek | Ric ( M ) | ≤ Λ , injrad ( M ) ≥ ε , diam ( M ) ≤ D,
⇒ locally symmetric Local symmetry = Symmetry gaps in Theorem (vL, 2014) Riemannian geometry There exists C ( n , Λ , ε, D ) such that if M n is as in the and minimal orbifolds conjecture, and Wouter van Limbeek | Ric ( M ) | ≤ Λ , injrad ( M ) ≥ ε , diam ( M ) ≤ D, then either [ Isom ( � M , � g ) : π 1 ( M )] ≤ C, or ( M , g ) is isometric to a locally symmetric space.
Proof Suppose there is no bound on [ Isom ( � Symmetry g ) : π 1 ( M )]. M , � gaps in Riemannian geometry and minimal orbifolds Wouter van Limbeek
Proof Suppose there is no bound on [ Isom ( � Symmetry g ) : π 1 ( M )]. M , � gaps in Riemannian geometry Choose g n such that [ Isom ( � and minimal M , � g n ) : π 1 ( M ) ] → ∞ . orbifolds � �� � � �� � Wouter van G n Γ Limbeek
Proof Suppose there is no bound on [ Isom ( � Symmetry g ) : π 1 ( M )]. M , � gaps in Riemannian geometry Choose g n such that [ Isom ( � and minimal M , � g n ) : π 1 ( M ) ] → ∞ . orbifolds � �� � � �� � Wouter van G n Γ Limbeek C 1 → g . Set G := Isom ( � ∃ g : g n − g ). M , � Easy facts: G is a Lie group, possibly with infinitely many components. Γ ⊆ G is a cocompact lattice.
Proof Suppose there is no bound on [ Isom ( � Symmetry g ) : π 1 ( M )]. M , � gaps in Riemannian geometry Choose g n such that [ Isom ( � and minimal M , � g n ) : π 1 ( M ) ] → ∞ . orbifolds � �� � � �� � Wouter van G n Γ Limbeek C 1 → g . Set G := Isom ( � ∃ g : g n − g ). M , � Easy facts: G is a Lie group, possibly with infinitely many components. Γ ⊆ G is a cocompact lattice. Show: [ G n : Γ] → ∞ = ⇒ [ G : Γ] = ∞ .
Proof Suppose there is no bound on [ Isom ( � Symmetry g ) : π 1 ( M )]. M , � gaps in Riemannian geometry Choose g n such that [ Isom ( � and minimal M , � g n ) : π 1 ( M ) ] → ∞ . orbifolds � �� � � �� � Wouter van G n Γ Limbeek C 1 → g . Set G := Isom ( � ∃ g : g n − g ). M , � Easy facts: G is a Lie group, possibly with infinitely many components. Γ ⊆ G is a cocompact lattice. Show: [ G n : Γ] → ∞ = ⇒ [ G : Γ] = ∞ . ⇒ G 0 � = 1 where G 0 is the connected component of = the identity.
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