Hyperbolic Orbifolds Of Small Volume
Hyperbolic orbifolds of small volumeMikhail BelolipetskyIMPAFriday, August 15. 16:00–16:452014 Seoul ICMhttp://icm2014.org/(IL5.2)
Volume in hyperbolic geometryH n – the hyperbolic n-space(e.g. the upper half space with the hyperbolic metric ds2 Isom(H n ) – the group of isometries of H n .dw2).y2
Volume in hyperbolic geometryH n – the hyperbolic n-space(e.g. the upper half space with the hyperbolic metric ds2 dw2).y2Isom(H n ) – the group of isometries of H n .Γ Isom(H n ), a discrete subgroup M H n /Γ is ahyperbolic n-orbifold.M is a manifold Γ is torsion free.
Volume in hyperbolic geometryH n – the hyperbolic n-space(e.g. the upper half space with the hyperbolic metric ds2 dw2).y2Isom(H n ) – the group of isometries of H n .Γ Isom(H n ), a discrete subgroup M H n /Γ is ahyperbolic n-orbifold.M is a manifold Γ is torsion free.We will discuss finite volume hyperbolic n-manifolds and orbifolds.
Volume in hyperbolic geometryFor n even:Vol(M ) Vol(Sn )·( 1)n/2 χ(M ) (Chern–Gauss–Bonnet Theorem)2
Volume in hyperbolic geometryFor n even:Vol(M ) Vol(Sn )·( 1)n/2 χ(M ) (Chern–Gauss–Bonnet Theorem)2For n 3 finite volume hyperbolic n-orbifolds are rigid(Mostow–Prasad rigidity) volume is a topological invariant.
Volume in hyperbolic geometryFor n even:Vol(M ) Vol(Sn )·( 1)n/2 χ(M ) (Chern–Gauss–Bonnet Theorem)2For n 3 finite volume hyperbolic n-orbifolds are rigid(Mostow–Prasad rigidity) volume is a topological invariant.If M is an oriented connected hyperbolic n-manifold,Vol(M ) νn kM k(Gromov–Thurston) volume is a measure of complexity.
Volume in hyperbolic geometry(Callahan–Dean–Weeks’1999)
Volume in hyperbolic geometryProblem 23. (Thurston, Bull. AMS, 1982) Show that volumes ofhyperbolic 3-manifolds are not all rationally related.
Volume in hyperbolic geometryProblem 23. (Thurston, Bull. AMS, 1982) Show that volumes ofhyperbolic 3-manifolds are not all rationally related.For even n the volumes are rationally related by the Gauss–Bonnettheorem.
Volume in hyperbolic geometryProblem 23. (Thurston, Bull. AMS, 1982) Show that volumes ofhyperbolic 3-manifolds are not all rationally related.For even n the volumes are rationally related by the Gauss–Bonnettheorem.The problem (restricted to arithmetic manifolds) is connected withdifficult open problems in number theory about rational independenceof certain Dedekind ζ -values.
Volume in hyperbolic geometryMinimal Volume Problem. Show that the volume of a hyperbolicn-orbifold is bounded below and find the minimal volume n-orbifoldsand manifolds.
Volume in hyperbolic geometryMinimal Volume Problem. Show that the volume of a hyperbolicn-orbifold is bounded below and find the minimal volume n-orbifoldsand manifolds.I(Siegel, 1945) Raised the problem and solved it for n 2.
Volume in hyperbolic geometryMinimal Volume Problem. Show that the volume of a hyperbolicn-orbifold is bounded below and find the minimal volume n-orbifoldsand manifolds.I(Siegel, 1945) Raised the problem and solved it for n 2.I(Kazhdan–Margulis, 1968) Proved the existence of the lowerbound in general.
Volume in hyperbolic geometryMinimal Volume Problem. Show that the volume of a hyperbolicn-orbifold is bounded below and find the minimal volume n-orbifoldsand manifolds.I(Siegel, 1945) Raised the problem and solved it for n 2.I(Kazhdan–Margulis, 1968) Proved the existence of the lowerbound in general.I(B., B.–Emery) Minimal volume arithmetic hyperbolicn-orbifolds for n 4.
Arithmeticity and volume: ExampleH 2 – the hyperbolic plane withthe Poincaré metric.Isom (H 2 ) PSL(2, R).Γ PSL(2, Z) PSL(2, R),a discrete subgroup.Γ acts on hyperbolic planewith O H 2 /Γ.
Arithmeticity and volume: ExampleH 2 – the hyperbolic plane withthe Poincaré metric.Isom (H 2 ) PSL(2, R).Γ PSL(2, Z) PSL(2, R),a discrete subgroup.Γ acts on hyperbolic planewith O H 2 /Γ.
Arithmeticity and volume: ExampleH 2 – the hyperbolic plane withthe Poincaré metric.Isom (H 2 ) PSL(2, R).Γ PSL(2, Z) PSL(2, R),a discrete subgroup.Γ acts on hyperbolic planewith O H 2 /Γ.ZZVol(O) F dx dy 2π χ(O)y21p3π 4π ζ ( 1) . π primes # PSL2 (Fp )3
Arithmeticity and volume: DefinitionsLet G be an algebraic group defined over a number field k.Let P (Pv )v Vf a collection of parahoric subgroups Pv G(kv ),where v runs through all finite places of k and kv denotes thenon-archimedean completion of the field. The family P is calledcoherent if v Vf Pv is an open subgroup of the finite adèle groupG(Af (k)). The groupΛ G(k) Pvv Vfis called the principal arithmetic subgroup of G(k) associated to P.
Arithmeticity and volume: DefinitionsLet G be an algebraic group defined over a number field k.Let P (Pv )v Vf a collection of parahoric subgroups Pv G(kv ),where v runs through all finite places of k and kv denotes thenon-archimedean completion of the field. The family P is calledcoherent if v Vf Pv is an open subgroup of the finite adèle groupG(Af (k)). The groupΛ G(k) Pvv Vfis called the principal arithmetic subgroup of G(k) associated to P.Example. SLn (Z) SLn (Q) p prime SLn (Zp ),
Arithmeticity and volume: DefinitionsLet G be an algebraic group defined over a number field k.Let P (Pv )v Vf a collection of parahoric subgroups Pv G(kv ),where v runs through all finite places of k and kv denotes thenon-archimedean completion of the field. The family P is calledcoherent if v Vf Pv is an open subgroup of the finite adèle groupG(Af (k)). The groupΛ G(k) Pvv Vfis called the principal arithmetic subgroup of G(k) associated to P.Example. SLn (Z) SLn (Q) p prime SLn (Zp ),Every maximal arithmetic subgroup is a normalizer of a principalarithmetic subgroup.
Groups versus coversIf Γ1 Γ0 , thenO1 H n /Γ1 coverO0 H n /Γ0
Groups versus coversIf Γ1 Γ0 , thenO1 H n /Γ1 coverO0 H n /Γ0Corollary. Minimal volume orbifolds correspond to maximal discretesubgroups.
Arithmeticity and volumeBorel–Harish-Chandra Theorem. Arithmetic subgroups are discreteand have finite covolume.
Arithmeticity and volumeBorel–Harish-Chandra Theorem. Arithmetic subgroups are discreteand have finite covolume.The volume of G/Γ can be computed using volume formulas:
Arithmeticity and volumeBorel–Harish-Chandra Theorem. Arithmetic subgroups are discreteand have finite covolume.The volume of G/Γ can be computed using volume formulas:IG. Harder, A Gauss–Bonnet formula for discrete arithmeticallydefined groups (Ann. Sci. École Norm. Sup., 1971)IA. Borel, Commensurability classes and volumes of hyperbolic3-manifolds (Ann. Scuola Norm. Sup. Pisa Cl. Sci., 1981)IG. Prasad, Volumes of S-arithmetic quotients of semi-simplegroups (Inst. Hautes Études Sci. Publ. Math., 1989)IB. Gross, On the motive of a reductive group(Invent. Math., 1997)
Arithmeticity and volumeBorel–Harish-Chandra Theorem. Arithmetic subgroups are discreteand have finite covolume.The volume of G/Γ can be computed using volume formulas:O H 2 / PSL(2, Z)Vol(O) 1π3p primes # PSL2 (Fp ) 4π ζ ( 1)
Results about minimal volumeH PO(n, 1) Isom (H n )
Results about minimal volumeH PO(n, 1) Isom (H n )Theorem 1. (B.’2004, B.–Emery’2012) For every dimension n 4there exists a unique cocompact arithmetic subgroupΓn0 H of the smallest covolume. It is defined over k0 Q[ 5] and hasVol(H n /Γn0 ) ωc (n).
Results about minimal volumeH PO(n, 1) Isom (H n )Theorem 1. (B.’2004, B.–Emery’2012) For every dimension n 4there exists a unique cocompact arithmetic subgroupΓn0 H of the smallest covolume. It is defined over k0 Q[ 5] and hasVol(H n /Γn0 ) ωc (n).Theorem 2. (B.’2004, B.–Emery’2012) For every dimension n 4there exists a unique non-cocompact arithmetic subgroup Γn1 H ofthe smallest covolume. It is defined over k1 Q and hasVol(H n /Γn1 ) ωnc (n).
n 2r, r even:2ωc (n) 4 · 5r r/2 · (2π)r r (2i 1)!2 (2π)4i ζk0 (2i);(2r 1)!!i 1n 2r, r odd:ωc (n) 2 · 5r2 r/2· (2π)r · (4r 1) r (2i 1)!2 (2π)4i ζk0 (2i);(2r 1)!!i 1(B.’2004)n 2r 1:ωc (n) 5r2 r/2· 11r 1/2 · (r 1)!L 0 k0(r)22r 1 π rr 1(2i 1)!2 (2π)4i ζk0 (2i),i 1 where k0 Q[ 5] and l0 is the quartic field with a definingpolynomial x4 x3 2x 1.(B.–Emery’2012)
n 2r, r even:2ωc (n) 4 · 5r r/2 · (2π)r r (2i 1)!2 (2π)4i ζk0 (2i);(2r 1)!!i 1n 2r, r odd:ωc (n) 2 · 5r2 r/2· (2π)r · (4r 1) r (2i 1)!2 (2π)4i ζk0 (2i);(2r 1)!!i 1(B.’2004)n 2r 1:ωc (n) 5r2 r/2· 11r 1/2 · (r 1)!L 0 k0(r)22r 1 π rr 1(2i 1)!2 (2π)4i ζk0 (2i),i 1 where k0 Q[ 5] and l0 is the quartic field with a definingpolynomial x4 x3 2x 1.(B.–Emery’2012)
n 2r, r 0, 1 (mod 4):ωnc (n) 4 · (2π)r r (2i 1)!ζ (2i);2i(2r 1)!! i 1 (2π)n 2r, r 2, 3 (mod 4):ωnc (n) 2 · (2r 1) · (2π)r r (2i 1)! (2π)2i ζ (2i);(2r 1)!!i 1(B.)n 2r 1, r even:ωnc (n) 3r 1/2L (r)2r 1 1 Q (2i 1)!ζ (2i), where 1 Q[ 3];2ii 1 (2π)r 1 n 2r 1, r 1 (mod 4):ωnc (n) 1r 1(2i 1)!ζ (2i);2ii 1 (2π)ζ (r) (2r 1)(2r 1 1)ζ (r)3 · 2r 1r 12r 2n 2r 1, r 3 (mod 4):ωnc (n) (2i 1)!ζ (2i);2ii 1 (2π) (B.–Emery)
Proofs useIPrasad’s volume formulaIGalois cohomology of algebraic groupsIBruhat–Tits theoryIBounds for discriminants and class numbers (Odlyzko bounds,Brauer–Siegel theorem, Zimmert’s bound for regulator)
Growth of minimal volume
CorollariesIThe minimal volume compact/non-compact arithmetichyperbolic n-orbifold in any dimension n is unique.
CorollariesIThe minimal volume compact/non-compact arithmetichyperbolic n-orbifold in any dimension n is unique.IThe values ωc (n), ωnc (n), and ωc (n)/ωnc (n) growsuper-exponentially.
CorollariesIThe minimal volume compact/non-compact arithmetichyperbolic n-orbifold in any dimension n is unique.IThe values ωc (n), ωnc (n), and ωc (n)/ωnc (n) growsuper-exponentially.IFor n 2r 12 the compact arithmetic manifolds have χ 2(in fact, this is true for all even n 6 — Emery’2014).
CorollariesIThe minimal volume compact/non-compact arithmetichyperbolic n-orbifold in any dimension n is unique.IThe values ωc (n), ωnc (n), and ωc (n)/ωnc (n) growsuper-exponentially.IFor n 2r 12 the compact arithmetic manifolds have χ 2(in fact, this is true for all even n 6 — Emery’2014).IFor n 5 we have ωc (n) ωnc (n) (“compact open”).
CorollariesConjecture. (B.–Emery) Let M be a compact hyperbolic manifoldof dimension n 6 3. Then there exists a noncompact hyperbolicn-manifold N whose volume is smaller than the volume of M .
CorollariesConjecture. (B.–Emery) Let M be a compact hyperbolic manifoldof dimension n 6 3. Then there exists a noncompact hyperbolicn-manifold N whose volume is smaller than the volume of M .The conjecture is true forn 2 – easyn 4 – follows from Ratcliffe–Tschantz’2000n 6 – follows from Everitt–Ratcliffe-Tschantz’2012arithmetic manifolds of dimension n 30 (B.–Emery’2013)
Minimal volume without arithmeticityLemma. (Margulis) For every dimension n there is a constantµ µn 0 such that for every discrete group Γ Isom(H n ) andevery x H n , the groupΓµ (x) hγ Γ dist(x, γ(x)) 6 µihas an abelian subgroup of finite index.
Minimal volume without arithmeticityLemma. (Margulis) For every dimension n there is a constantµ µn 0 such that for every discrete group Γ Isom(H n ) andevery x H n , the groupΓµ (x) hγ Γ dist(x, γ(x)) 6 µihas an abelian subgroup of finite index.Theorem. (Gelander) Given a hyperbolic n-orbifold O n , we haveVol(O n ) 2v(0.25ε)2,v(1.25ε)ε min{µn, 1}.10
Minimal volume without arithmeticityLemma. (Margulis) For every dimension n there is a constantµ µn 0 such that for every discrete group Γ Isom(H n ) andevery x H n , the groupΓµ (x) hγ Γ dist(x, γ(x)) 6 µihas an abelian subgroup of finite index.Theorem. (Gelander) Given a hyperbolic n-orbifold O n , we haveVol(O n ) 2v(0.25ε)2,v(1.25ε)ε min{µn, 1}.10Proposition. There exists a constant C 0 such that µn 6C .n
Minimal volume without arithmeticityCorollary. The lower bound for the volume decreasessuper-exponentially with n.
Minimal volume without arithmeticityCorollary. The lower bound for the volume decreasessuper-exponentially with n.Remark. The same is true for the bound of Adeboye–Wei obtainedby quantifying the proof of the Kazhdan–Margulis Theorem.
Minimal volume without arithmeticityCorollary. The lower bound for the volume decreasessuper-exponentially with n.Remark. The same is true for the bound of Adeboye–Wei obtainedby quantifying the proof of the Kazhdan–Margulis Theorem.Conjecture. The minimal volume hyperbolic n-orbifold (manifold) isarithmetic.
Minimal volume without arithmeticityCorollary. The lower bound for the volume decreasessuper-exponentially with n.Remark. The same is true for the bound of Adeboye–Wei obtainedby quantifying the proof of the Kazhdan–Margulis Theorem.Conjecture. The minimal volume hyperbolic n-orbifold (manifold) isarithmetic.It follows from the conjecture that we expect the minimal volume togrow super-exponentially but so far we can prove onlysuper-exponentially decreasing bounds!
Volume in hyperbolic geometry H n - the hyperbolic n-space (e.g. the upper half space with the hyperbolic metric ds2 dw2 y2). Isom(H n) - the group of isometries of H n. G Isom(H n), a discrete subgroup )M H n G is a hyperbolic n-orbifold. M is a manifold ()G is torsion free. We will discuss finite volume hyperbolic n-manifolds and .
geodesic distance is the arc-length of the unique hyperbolic line joining x and y and equals the hyperbolic distance d(x;y). 1.3 The Conformal Model of Hyperbolic Geometry Distorting the projective model leads to a new model of hyperbolic geometry with some other special metric properties: Let H2 be the interior of the unit circle and de ne .
equal to ˇ. In hyperbolic geometry, 0 ˇ. In spherical geometry, ˇ . Figure 1. L to R, Triangles in Euclidean, Hyperbolic, and Spherical Geometries 1.1. The Hyperbolic Plane H. The majority of 3-manifolds admit a hyperbolic struc-ture [Thurston], so we shall focus primarily on the hyperbolic geometry, starting with the hyperbolic plane, H.
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1 Hyperbolic space and its isometries 1 1.1 Möbius transformations 1 1.2 Hyperbolic geometry 6 1.2.1 The hyperbolic plane 8 1.2.2 Hyperbolic space 8 1.3 The circle or sphere at infinity 12 1.4 Gaussian curvature 16 1.5 Further properties of Möbius transformations 19 1.5.1 Commutativity 19 1.5.2 Isometric circles and planes 20 1.5.3 Trace .
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One difficulty for defining the hyperbolic CVT is how to well de-fine the centroid of a given region in 2D hyperbolic space. In this paper, we extend the model centroid [Galperin 1993] to define the centroid of a Voronoi cell in 2D hyperbolic space. Another chal-lenge is how to effectively and efficiently compute the hyperbolic CVT.
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