On Symmetry of Flat Manifolds Rafał Lutowski Institute of Mathematics, University of Gda´ nsk Conference on Algebraic Topology CAT’09 July 6-11, 2009 Warsaw Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 1 / 29
Outline Introduction 1 Fundamental Groups of Flat Manifolds Affine Self Equivalences of Flat Manifolds Examples Flat Manifold with Odd-Order Group of Symmetries 2 Construction Outer Automorphism Group Direct Products of Centerless Bieberbach Groups 3 (Outer) Automorphism Groups Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 2 / 29
Introduction Outline Introduction 1 Fundamental Groups of Flat Manifolds Affine Self Equivalences of Flat Manifolds Examples Flat Manifold with Odd-Order Group of Symmetries 2 Construction Outer Automorphism Group Direct Products of Centerless Bieberbach Groups 3 (Outer) Automorphism Groups Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 3 / 29
Introduction Fundamental Groups of Flat Manifolds Flat Manifolds and Bieberbach Groups X – compact, connected, flat Riemannian manifold (flat manifold for short). Γ = π 1 ( X ) – fundamental group of X – Bieberbach group. X is isometric to R n / Γ . Γ determines X up to affine equivalence. Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 4 / 29
Introduction Fundamental Groups of Flat Manifolds Abstract Definition of Bieberbach Groups Definition Bieberbach group is a torsion-free group defined by a short exact sequence 0 − → M − → Γ − → G − → 1 . G – finite group (holonomy group of Γ ). M – faithful G -lattice, i.e. faithful and free Z G -module, finitely generated as an abelian group. Element α ∈ H 2 ( G, M ) corresponding to the above extension is special, i.e. res G H α � = 0 for every non-trivial subgroup H of G . Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 5 / 29
Introduction Affine Self Equivalences of Flat Manifolds Group of Affinities Aff( X ) – group of affine self equivalences of X . ◮ Aff( X ) is a Lie group. Aff 0 ( X ) – identity component of Aff( X ) . ◮ Aff 0 ( X ) is a torus. ◮ Dimension of Aff 0 ( X ) equals b 1 ( X ) – the first Betti number of X ( b 1 ( X ) = rk H 0 ( G, M ) ). Theorem (Charlap, Vasquez 1973) Aff( X ) / Aff 0 ( X ) ∼ = Out(Γ) Corollary Aff( X ) is finite iff b 1 ( X ) = 0 and Out(Γ) is finite. Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 6 / 29
Introduction Affine Self Equivalences of Flat Manifolds Problem Problem (Szczepa´ nski 2006) Which finite groups occur as outer automorphism groups of Bieberbach groups with trivial center. Theorem (Belolipetsky, Lubotzky 2005) For every n ≥ 2 and every finite group G there exist infinitely many compact n -dimensional hyperbolic manifolds M with Isom( M ) ∼ = G . Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 7 / 29
Introduction Examples Calculating Out(Γ) Theorem (Charlap, Vasquez 1973) Out(Γ) fits into short exact sequence → H 1 ( G, M ) − 0 − → Out(Γ) − → N α /G − → 1 . N α – stabilizer of α ∈ H 2 ( G, M ) under the action of N Aut( M ) ( G ) defined by n ∗ a ( g 1 , g 2 ) = n · a ( n − 1 g 1 n, n − 1 g 2 n ) . Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 8 / 29
Introduction Examples Finite Groups of Affinities of Flat Manifolds (Szczepa´ nski, Hiss 1997) ◮ C 2 – two flat manifolds. ◮ C 2 × ( C 2 ≀ F ) , where F ⊂ S 2 k +1 is cyclic group generated by the cycle (1 , 2 , . . . , 2 k + 1) , k ≥ 2 . (Waldmüller 2003) ◮ A flat manifold with no symmetries. (Lutowski, PhD) ◮ C k 2 , k ≥ 2 . Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 9 / 29
Flat Manifold with Odd-Order Group of Symmetries Outline Introduction 1 Fundamental Groups of Flat Manifolds Affine Self Equivalences of Flat Manifolds Examples Flat Manifold with Odd-Order Group of Symmetries 2 Construction Outer Automorphism Group Direct Products of Centerless Bieberbach Groups 3 (Outer) Automorphism Groups Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 10 / 29
Flat Manifold with Odd-Order Group of Symmetries Lattice Basis Definition Bieberbach group is a torsion-free group defined by a short exact sequence → Z n − 0 − → Γ − → G − → 1 . G ֒ → GL n ( Z ) – integral representation of G . G acts on Z n by matrix multiplication. Element α ∈ H 2 ( G, Z n ) corresponding to the above extension is special. Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 11 / 29
Flat Manifold with Odd-Order Group of Symmetries Special Element of H 2 ( G, M ) α ∈ H 2 ( G, Z n ) is special iff res G H α � = 0 for every 1 � = H < G . By the transitivity of restriction – enough to check subgroups of prime order. By the action ’ ∗ ’ of normalizer – enough to check conjugacy classes of such groups. Since H 2 ( G, Z n ) is hard to compute, we use an isomorphic group H 1 ( G, Q n / Z n ) . Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 12 / 29
Flat Manifold with Odd-Order Group of Symmetries Construction Holonomy Group G = M 11 – Mathieu group on 11 letters. | G | = 7920 = 2 4 · 3 2 · 5 · 11 . G has a presentation G = � a, b | a 2 , b 4 , ( ab ) 11 , ( ab 2 ) 6 , ababab − 1 abab 2 ab − 1 abab − 1 ab − 1 � . Representatives of conjugacy classes of G : ◮ Order 2: � a � , ◮ Order 3: � ( ab 2 ) 2 � , ◮ Order 5: Sylow subgroups, ◮ Order 11: Sylow subgroups. Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 13 / 29
Flat Manifold with Odd-Order Group of Symmetries Construction The Lattice – Definition The lattice is given by integral representation of G . M 1 , M 3 , M 4 – representation from Waldmüller’s example of degree 20,44,45 respectively. M 3 – sublattice of index 3 of Waldmüller’s lattice of degree 32, given by the orbit of the vector (2 , 1 , . . . , 1 ) . � �� � 32 M := M 1 ⊕ . . . ⊕ M 4 . Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 14 / 29
Flat Manifold with Odd-Order Group of Symmetries Construction The Lattice – Properties H 1 ( G, M i ) H 2 ( G, M i ) i Degree C -irr |� α i �| | H i | 1 20 No 0 C 6 6 3 2 32 No C 3 C 5 5 5 0 C 6 3 44 Yes 6 2 4 45 Yes 0 C 11 11 11 H 1 ( G, M ) = � 4 i =1 H 1 ( G, M i ) = C 3 . Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 15 / 29
Flat Manifold with Odd-Order Group of Symmetries Construction Torsion-Free Extension Proposition Extension Γ of M by G defined by α := α 1 ⊕ . . . ⊕ α 4 ∈ H 2 ( G, M ) is torsion-free. Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 16 / 29
Flat Manifold with Odd-Order Group of Symmetries Outer Automorphism Group Next Step in Calculating Out(Γ) Recall short exact sequence → H 1 ( G, M ) − 0 − → Out(Γ) − → N α /G − → 1 . H 1 ( G, M ) = C 3 . Next step: calculate N α . Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 17 / 29
Flat Manifold with Odd-Order Group of Symmetries Outer Automorphism Group Calculation of Stabilizer Step 1: Centralizer C Aut( M ) ( G ) = C Aut( M 1 ) ( G ) × . . . × C Aut( M 4 ) ( G ) . M 3 , M 4 – absolutely irreducible, thus C Aut( M i ) ( G ) = �− 1 � , k = 3 , 4 . For k = 1 , 2 : C Aut( M k ) ( G ) = U (End Z G ( M k )) . We have: = Z [ √− 2] , ◮ End Z G ( M 1 ) ∼ ] ⊂ Z [ √− 11 , 1 = Z [ 3 √− 11 − 1 ◮ End Z G ( M 2 ) ∼ 2 ] . 2 U (End Z G ( M k )) = �− 1 � , k = 1 , 2 . Corollary C Aut( M ) ( G ) α = 1 . Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 18 / 29
Flat Manifold with Odd-Order Group of Symmetries Outer Automorphism Group Calculation of Stabilizer Step 2: Normalizer Since Out( G ) = 1 , we have N Aut( M ) ( G ) = G · C Aut( M ) ( G ) . Corollary N Aut( M ) ( G ) α = G. Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 19 / 29
Flat Manifold with Odd-Order Group of Symmetries Outer Automorphism Group Flat Manifold with Odd-Order Group of Symmetries Theorem If X is a manifold with fundamental group Γ , then Aff( X ) ∼ = C 3 . Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 20 / 29
Flat Manifold with Odd-Order Group of Symmetries Outer Automorphism Group Further properties of Γ Aut(Γ) is a Bieberbach group. Out(Aut(Γ)) = 1 . � Γ: Γ ′ � = 3 ∧ Out(Γ ′ ) ∼ = C 2 ∃ Γ ′ ⊳ Γ 3 . Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 21 / 29
Direct Products of Centerless Bieberbach Groups Outline Introduction 1 Fundamental Groups of Flat Manifolds Affine Self Equivalences of Flat Manifolds Examples Flat Manifold with Odd-Order Group of Symmetries 2 Construction Outer Automorphism Group Direct Products of Centerless Bieberbach Groups 3 (Outer) Automorphism Groups Rafał Lutowski (University of Gda´ nsk) On Symmetry of Flat Manifolds CAT’09 22 / 29
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