computing fundamental domains for arithmetic kleinian...

Hyperbolic geometryQuaternion algebras

AlgorithmsExamples

Computing fundamental domainsfor arithmetic Kleinian groups

Aurel Page

August 24, 2010

Aurel Page Computing fundamental domains for Kleinian groups

AlgorithmsExamples

1 Hyperbolic geometryAction of PSL2(C) on the upper half-spaceKleinian groups and fundamental domainsFace pairing

2 Quaternion algebrasQuaternion algebras and splittingOrders and arithmetic Kleinian groupsApplications

3 AlgorithmsThe unit ball and isometric spheresComputing the volumeThe absolute reduced normThe reduction algorithm

4 ExamplesBianchi groupsCocompact groups

AlgorithmsExamples

Action of PSL2(C) on the upper half-spaceKleinian groups and fundamental domainsFace pairing

Definition

The upper half-space is the set H3 = C× R>0 with the metric

ds2 =dx2 + dy 2 + dt2

where (z , t) ∈ H3, z = x + iy and t > 0. The set C = P1(C) isthe sphere at infinity, and the completed upper half-spaceis H3 = H3 ∪ C.

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Definition

ds2 =dx2 + dy 2 + dt2

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Definition

ds2 =dx2 + dy 2 + dt2

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Definition (Action of PSL2(C) on H3)

Let γ = ±(

a bc d

)∈ PSL2(C). Using H3 ∼= C + R>0j ⊂ H, we

defineγ · z = (az + b)(cz + d)−1.

AlgorithmsExamples

Definition (Action of PSL2(C) on H3)

Let γ = ±(

a bc d

)∈ PSL2(C). Using H3 ∼= C + R>0j ⊂ H, we

defineγ · z = (az + b)(cz + d)−1.

AlgorithmsExamples

Classification of elements γ 6= 1 ∈ PSL2(C):

tr(γ) ∈ C \ [−2, 2]: γ is loxodromic.

tr(γ) ∈ (−2, 2): γ is elliptic.

tr(γ) = ±2: γ is parabolic.

AlgorithmsExamples

Definition

A Kleinian group is a discrete subgroup of PSL2(C).

Example

If F ⊂ C is a quadratic imaginary field and ZF is the ring ofintegers of F , then the Bianchi group PSL2(ZF ) is a Kleiniangroup.

AlgorithmsExamples

Definition

A Kleinian group is a discrete subgroup of PSL2(C).

Example

If F ⊂ C is a quadratic imaginary field and ZF is the ring ofintegers of F , then the Bianchi group PSL2(ZF ) is a Kleiniangroup.

AlgorithmsExamples

Definition

Let Γ be a Kleinian group. A fundamental domain for Γ is an openconnected subset F of H3 such that

(i)⋃γ∈Γ γ · F = H3;

(ii) For all γ ∈ Γ \ {1}, F ∩ γ · F = ∅;(iii) Vol(∂F) = 0.

A fundamental domain that is a polyhedron is a fundamentalpolyhedron, it is finite if it has finitely many faces.If Γ admits afundamental domain with compact closure, then we say Γ iscocompact. If Γ admits a fundamental domain F with finitevolume, then we say Γ has finite covolume and we let

Covol(Γ) = Vol(F).

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Definition

(i)⋃γ∈Γ γ · F = H3;

Covol(Γ) = Vol(F).

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Definition

(i)⋃γ∈Γ γ · F = H3;

Covol(Γ) = Vol(F).

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Definition

(i)⋃γ∈Γ γ · F = H3;

Covol(Γ) = Vol(F).

AlgorithmsExamples

Given a Kleinian group Γ, compute a fundamental domain for Γ.

AlgorithmsExamples

Proposition

Let Γ be a Kleinian group. Let p ∈ H3 be a point with trivialstabilizer in Γ. Then the set

Dp(Γ) = {x ∈ H3 | for all γ ∈ Γ \ {1}, d(x , p) < d(γ · x , p)}

is a convex fundamental polyhedron for Γ. If it has finite covolume,then it is finite.

Definition

The domain Dp(Γ) is a Dirichlet domain for Γ.

AlgorithmsExamples

Proposition

Let Γ be a Kleinian group. Let p ∈ H3 be a point with trivialstabilizer in Γ. Then the set

Dp(Γ) = {x ∈ H3 | for all γ ∈ Γ \ {1}, d(x , p) < d(γ · x , p)}

is a convex fundamental polyhedron for Γ. If it has finite covolume,then it is finite.

Definition

The domain Dp(Γ) is a Dirichlet domain for Γ.

AlgorithmsExamples

Given a Kleinian group Γ, compute a Dirichlet domain for Γ.

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Definition

Let F be a Dirichlet domain for a Kleinian group Γ, and F the setof faces of F . A face pairing is a map ·∗ × g : F → F × Γ whichassigns to every face f a face f ∗ and a element g(f ) ∈ Γ s.t.

(a) g(f ) · f = f ∗;

(b) ·∗ : F → F is an involution;

The elements g(f ) where f is a face of F are the pairingtransformations.

Proposition

Any Dirichlet domain admits a face pairing. The pairingtransformations are generators for the group Γ

AlgorithmsExamples

Definition

(a) g(f ) · f = f ∗;

Proposition

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Definition

(a) g(f ) · f = f ∗;

Proposition

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Definition

(a) g(f ) · f = f ∗;

Proposition

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Definition

(a) g(f ) · f = f ∗;

Proposition

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Definition (Cycles)

Let F be a finite Dirichlet domain for a Kleinian group Γ. Defineby induction:

Pick e1 an edge of F ;

e1 = f ∩ f1;

g1 = g(f1);

ei+1 = gi · ei ;ei+1 = fi+1 ∩ f ∗i ;

gi+1 = g(fi+1).

The sequence (ei ) is periodic; let m be its period.C = (e1, . . . , em) is a cycle of edges. The cycle transformation ate1 is h = gmgm−1 . . . g1.

AlgorithmsExamples

Definition (Cycles)

e1 = f ∩ f1;

g1 = g(f1);

ei+1 = gi · ei ;ei+1 = fi+1 ∩ f ∗i ;

gi+1 = g(fi+1).

AlgorithmsExamples

Definition (Cycles)

e1 = f ∩ f1;

g1 = g(f1);

ei+1 = gi · ei ;ei+1 = fi+1 ∩ f ∗i ;

gi+1 = g(fi+1).

AlgorithmsExamples

Definition (Cycles)

e1 = f ∩ f1;

g1 = g(f1);

ei+1 = gi · ei ;ei+1 = fi+1 ∩ f ∗i ;

gi+1 = g(fi+1).

AlgorithmsExamples

Definition (Cycles)

e1 = f ∩ f1;

g1 = g(f1);

ei+1 = gi · ei ;ei+1 = fi+1 ∩ f ∗i ;

gi+1 = g(fi+1).

AlgorithmsExamples

Definition (Cycles)

e1 = f ∩ f1;

g1 = g(f1);

ei+1 = gi · ei ;ei+1 = fi+1 ∩ f ∗i ;

gi+1 = g(fi+1).

AlgorithmsExamples

Definition (Cycles)

e1 = f ∩ f1;

g1 = g(f1);

ei+1 = gi · ei ;ei+1 = fi+1 ∩ f ∗i ;

gi+1 = g(fi+1).

AlgorithmsExamples

Definition (Cycles)

e1 = f ∩ f1;

g1 = g(f1);

ei+1 = gi · ei ;ei+1 = fi+1 ∩ f ∗i ;

gi+1 = g(fi+1).

AlgorithmsExamples

Proposition

Let F be a finite Dirichlet domain for a Kleinian group Γ.

If a face f has f ∗ = f , then we have the reflectionrelation g(f )2 = 1;

For all edges e with cycle transformation h and for all x ∈ ewe have h · x = x; h satisfies the cycle relation hν = 1.

The reflection relations and the cycle relations form a complete setof relations for Γ.

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Proposition

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Proposition

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Proposition

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Given a Kleinian group Γ, compute a Dirichlet domain for Γ with aface pairing, and a presentation for Γ.

Compute the inverse isomorphism with the abstract finitelypresented group.

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Given a Kleinian group Γ, compute a Dirichlet domain for Γ with aface pairing, and a presentation for Γ.

Compute the inverse isomorphism with the abstract finitelypresented group.

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Quaternion algebras and splittingOrders and arithmetic Kleinian groupsApplications

Definition

Let F be a field with char F 6= 2 and a, b ∈ F×. An F -algebraadmitting a presentation of the form

〈 i , j | i2 = a, j2 = b, ij = −ji 〉

is the quaternion algebra(a,bF

)over F .

Example

The matrix algebra is M2(F ) ∼=(

)via the homomorphism(

1 00 −1

)7→ i

(0 11 0

)7→ j .

AlgorithmsExamples

Definition

Let F be a field with char F 6= 2 and a, b ∈ F×. An F -algebraadmitting a presentation of the form

〈 i , j | i2 = a, j2 = b, ij = −ji 〉

is the quaternion algebra(a,bF

)over F .

Example

The matrix algebra is M2(F ) ∼=(

)via the homomorphism(

1 00 −1

)7→ i

(0 11 0

)7→ j .

AlgorithmsExamples

Definition

Let B =(a,bF

)be a quaternion algebra and

β = x + yi + zj + tij ∈ B. The conjugate, reduced trace andreduced norm of β are β = x + yi + zj + tij , trd(β) = β + β = 2xand nrd(β) = ββ = x2 − ay 2 − bz2 + abt2.

Example

In the matrix ring, the reduced trace is the usual trace and thereduced norm is the determinant.

AlgorithmsExamples

Definition

Let B =(a,bF

)be a quaternion algebra and

β = x + yi + zj + tij ∈ B. The conjugate, reduced trace andreduced norm of β are β = x + yi + zj + tij , trd(β) = β + β = 2xand nrd(β) = ββ = x2 − ay 2 − bz2 + abt2.

Example

In the matrix ring, the reduced trace is the usual trace and thereduced norm is the determinant.

AlgorithmsExamples

Definition

Let B =(a,bF

)be a quaternion algebra over a number field F . A

place v of F is split or ramified according as B ⊗F Fv =M2(Fv )or not, where Fv denotes the completion of F at v . The product ofall ramified primes p ⊂ ZF is the discriminant ∆B of B.

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Definition

Let F be a number field and ZF its ring of integers. Let B be aquaternion algebra over F . An order in B is a finitelygenerated ZF -submodule O ⊂ B with FO = B which is also asubring. We write O×1 the group of units in O with reducednorm 1.

Example

Let F and ZF be as above, a, b ∈ ZF \ {0} and B =(a,bF

). Then

the ZF -module O = ZF + ZF i + ZF j + ZF ij is an order in B andthe ZF -module M2(ZF ) is an order in M2(F ).

AlgorithmsExamples

Definition

Example

). Then

the ZF -module O = ZF + ZF i + ZF j + ZF ij is an order in B. andthe ZF -module M2(ZF ) is an order in M2(F ).

AlgorithmsExamples

Definition

Example

). Then

the ZF -module O = ZF + ZF i + ZF j + ZF ij is an order in B, andthe ZF -module M2(ZF ) is an order in M2(F ).

AlgorithmsExamples

Definition

Let F be a number field. We say F is quasi totally real or QTRif F has exactly one complex place. A Kleinian quaternion algebrais a quaternion algebra over a QTR number field, ramified at everyreal place.

Example

A quadratic imaginary field is a QTR number field. For anypositive cubefree integer d 6= 1, Q( 3

√d) is a QTR number field.

AlgorithmsExamples

Definition

Let F be a number field. We say F is quasi totally real or QTRif F has exactly one complex place. A Kleinian quaternion algebrais a quaternion algebra over a QTR number field, ramified at everyreal place.

Example

A quadratic imaginary field is a QTR number field. For anypositive cubefree integer d 6= 1, Q( 3

√d) is a QTR number field.

AlgorithmsExamples

Theorem

Let B be a Kleinian quaternion algebra over a QTR numberfield F . Then there is a discrete embedding

ρ : O×1 ↪→ SL2(C).

The Kleinian group Γ = ρ(O×1 )/± 1 has finite covolume and it iscocompact if and only if B is a division algebra.If B is not adivision algebra then the base field F of B is a quadratic imaginaryfield and B ∼=M2(F ).If O is maximal, then we have

Covol(Γ) =|∆F |3/2ζF (2)

∏p|∆B

(N(p)− 1)

(4π2)n−1

where ∆F is the discriminant of F , ζF is the Dedekind zetafunction of F and ∆B is the discriminant of B.

AlgorithmsExamples

Theorem

ρ : O×1 ↪→ SL2(C).

The Kleinian group Γ = ρ(O×1 )/± 1 has finite covolume. and it iscocompact if and only if B is a division algebra.If B is not adivision algebra then the base field F of B is a quadratic imaginaryfield and B ∼=M2(F ).If O is maximal, then we have

Covol(Γ) =|∆F |3/2ζF (2)

∏p|∆B

(N(p)− 1)

(4π2)n−1

AlgorithmsExamples

Theorem

ρ : O×1 ↪→ SL2(C).

The Kleinian group Γ = ρ(O×1 )/± 1 has finite covolume, and it iscocompact if and only if B is a division algebra.If B is not adivision algebra then the base field F of B is a quadratic imaginaryfield and B ∼=M2(F ).If O is maximal, then we have

Covol(Γ) =|∆F |3/2ζF (2)

∏p|∆B

(N(p)− 1)

(4π2)n−1

AlgorithmsExamples

Theorem

ρ : O×1 ↪→ SL2(C).

Covol(Γ) =|∆F |3/2ζF (2)

∏p|∆B

(N(p)− 1)

(4π2)n−1

AlgorithmsExamples

Theorem

ρ : O×1 ↪→ SL2(C).

Covol(Γ) =|∆F |3/2ζF (2)

∏p|∆B

(N(p)− 1)

(4π2)n−1

AlgorithmsExamples

An Kleinian group is arithmetic if it is commensurable withsome ρ(O×1 )/± 1 as in the previous theorem. For example, werecover the Bianchi groups by taking F a quadratic imaginary field,B =M2(F ) and O =M2(ZF ) so that O×1 = SL2(ZF ).

Given an arithmetic Kleinian group Γ, compute a Dirichlet domainfor Γ with a face pairing, and a presentation for Γ.

AlgorithmsExamples

An Kleinian group is arithmetic if it is commensurable withsome ρ(O×1 )/± 1 as in the previous theorem. For example, werecover the Bianchi groups by taking F a quadratic imaginary field,B =M2(F ) and O =M2(ZF ) so that O×1 = SL2(ZF ).

Given an arithmetic Kleinian group Γ, compute a Dirichlet domainfor Γ with a face pairing, and a presentation for Γ.

AlgorithmsExamples

Applications

Compute the structure of unit groups O× where O is an orderin a Kleinian quaternion algebra;

Compute the cohomology of arithmetic Kleinian groups withthe action of Hecke operators;

Study a large class of compact hyperbolic 3-manifolds.

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Applications

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Applications

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The unit ball and isometric spheresComputing the volumeThe absolute reduced normThe reduction algorithm

Definition

The unit ball B is the open ball of center 0 and radius 1in R3 ∼= C + Rj ⊂ H with the metric

ds2 =4(dx2 + dy 2 + dt2)

(1− |w |2)2

where w = (z , t) ∈ B, z = x + iy and |w |2 = x2 + y 2 + t2 ≤ 1.The sphere at infinity ∂B is the sphere of center 0 and radius 1.We let B = B ∪ ∂B be the closed ball of radius 1.

AlgorithmsExamples

Proposition

The map

{H3 −→ B

z 7−→ (z − j)(1− jz)−1

is a bijective isometry;

For all

w , z ∈ B, d(w , z) = cosh−1

(1 + 2

|w − z |2

(1− |w |2)(1− |z |2)

AlgorithmsExamples

Proposition

The map

{H3 −→ B

z 7−→ (z − j)(1− jz)−1

is a bijective isometry;

For all

w , z ∈ B, d(w , z) = cosh−1

(1 + 2

|w − z |2

(1− |w |2)(1− |z |2)

AlgorithmsExamples

For all w ∈ B, g =

(a bc d

)∈ SL2(C), let

g · w = η(g · η−1(w)).

Proposition

g · w = (Aw + B)(Cw + D)−1

whereA = a + d + (b − c)j , B = b + c + (a− d)j ,

C = c + b + (d − a)j , D = d + a + (c − b)j .

AlgorithmsExamples

For all w ∈ B, g =

(a bc d

)∈ SL2(C), let

g · w = η(g · η−1(w)).

Proposition

g · w = (Aw + B)(Cw + D)−1

whereA = a + d + (b − c)j , B = b + c + (a− d)j ,

C = c + b + (d − a)j , D = d + a + (c − b)j .

AlgorithmsExamples

Definitions

Suppose g ∈ SL2(C) does not fix 0 in B. Then let

I(g) = {w ∈ B | d(w , 0) = d(g · w , 0)};Ext(g) = {w ∈ B | d(w , 0) < d(g · w , 0)};Int(g) = {w ∈ B | d(w , 0) > d(g · w , 0)};

I(g) is the isometric sphere of g . For S ⊂ SL2(C) with no elementfixing 0, the exterior domain of S is

Ext(S) =⋂g∈S

Ext(g).

For S a Euclidean sphere, define Ext(S) (resp. Int(S)) to be theintersection of B and the exterior (resp. the interior) of the sphere.

AlgorithmsExamples

Proposition

Let g =

(a bc d

)∈ SL2(C) and A,B,C ,D as in the previous

proposition. Then g · 0 = 0 if and only if C = 0. If g does notfix 0, then I(g) is the intersection of B and the Euclidean sphere ofcenter −C−1D and radius 2/|C |, and we have Int(g) = Int(I(g))and Ext(g) = Ext(I(g)).

AlgorithmsExamples

Remark

If 0 ∈ B has a trivial stabilizer in the Kleinian group Γ,then wehave D0(Γ) = Ext(Γ \ {1}).

Let γ ∈ Γ and F = Ext(Γ \ {1}). Then γ · I(γ) = I(γ−1), and I(γ)contributes to the boundary of F if and only if I(γ−1) does.

Remark

The face pairing of an exterior domain is given by the isometricspheres.

AlgorithmsExamples

Remark

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Remark

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Given an arithmetic Kleinian group Γ, compute the exterior domainof Γ with the face pairing, and a presentation for Γ.

Algorithm

Enumerate the element of Γ in a finite set S until we have

Vol(Ext(S)) = Covol(Γ).

AlgorithmsExamples

Given an arithmetic Kleinian group Γ, compute the exterior domainof Γ with the face pairing, and a presentation for Γ.

Algorithm

Enumerate the element of Γ in a finite set S until we have

Vol(Ext(S)) = Covol(Γ).

AlgorithmsExamples

Proposition

The integral

−∫ θ

0ln |2 sin u| du

converges for θ ∈ R \ πZ and admits a continuous extension to R,which is odd and periodic with period π.

Definition

This extension is called the Lobachevsky function L(θ).

AlgorithmsExamples

Proposition

The integral

−∫ θ

0ln |2 sin u| du

converges for θ ∈ R \ πZ and admits a continuous extension to R,which is odd and periodic with period π.

Definition

This extension is called the Lobachevsky function L(θ).

AlgorithmsExamples

Proposition

The Lobachevsky function admits a power series expansion:

L(θ) = θ

(1− ln(2θ) +

∞∑n=1

22n|B2n|2n(2n + 1)!

where the Bn are the Bernoulli numbers defined by

ex − 1=∞∑n=0

AlgorithmsExamples

Proposition

Let Tα,γ be the tetrahedron in H3 with one vertex at ∞ and theother vertices A,B,C on the unit hemisphere such that theyproject vertically onto A,B ′,C ′ in C with A′ = 0 to form aEuclidean triangle, with angles π

2 at B ′ with and α at A′, and suchthat the angle along BC is γ. Then Tα,γ is unique up to isometryand

Vol(Tα,γ) =1

[L(α + γ) + L(α− γ) + 2L

(π2− α

Remark

Every polyhedron P can be decomposed in a way such that thevolume of P is a sum of volumes of tetrahedra Tα,γ .

AlgorithmsExamples

Proposition

Let Tα,γ be the tetrahedron in H3 with one vertex at ∞ and theother vertices A,B,C on the unit hemisphere such that theyproject vertically onto A,B ′,C ′ in C with A′ = 0 to form aEuclidean triangle, with angles π

2 at B ′ with and α at A′, and suchthat the angle along BC is γ. Then Tα,γ is unique up to isometryand

Vol(Tα,γ) =1

[L(α + γ) + L(α− γ) + 2L

(π2− α

Remark

Every polyhedron P can be decomposed in a way such that thevolume of P is a sum of volumes of tetrahedra Tα,γ .

AlgorithmsExamples

Definition

Let m =

(a bc d

)∈M2(C) and define invrad(m) = |C |2.

For γ ∈ SL2(C) with γ · 0 6= 0, let rad(γ) be the radius of I(γ).

Proposition

Suppose ρ : O×1 ↪→ SL2(C) is a discrete embedding. The absolutereduced norm Q : B → R defined by

Q(x) = invrad(ρ(x)) + trF/Q(nrd(x)) for all x ∈ B

gives O the structure of a lattice, and we have

for all x ∈ O×1 , Q(x) =4

rad(ρ(x))2+ n.

AlgorithmsExamples

Definition

Let m =

(a bc d

)∈M2(C) and define invrad(m) = |C |2.

For γ ∈ SL2(C) with γ · 0 6= 0, let rad(γ) be the radius of I(γ).

Proposition

Suppose ρ : O×1 ↪→ SL2(C) is a discrete embedding. The absolutereduced norm Q : B → R defined by

Q(x) = invrad(ρ(x)) + trF/Q(nrd(x)) for all x ∈ B

gives O the structure of a lattice, and we have

for all x ∈ O×1 , Q(x) =4

rad(ρ(x))2+ n.

AlgorithmsExamples

Definition

Let Γ be a Kleinian group and S ⊂ Γ. A point z ∈ B is S-reducedif for all g ∈ S , we have d(z , 0) ≤ d(g · z , 0), i.e. if z ∈ Ext(S).

Algorithm

Input: z ∈ B.Let z ′ = z . If d(g · z ′, 0) < d(z ′, 0) for some g ∈ S , thenset z ′ = g · z ′ and repeat.Output: z ′ ∈ B, S-reduced and δ ∈ 〈S〉 s.t. z ′ = δ · z .

AlgorithmsExamples

Definition

Let Γ be a Kleinian group and S ⊂ Γ. A point z ∈ B is S-reducedif for all g ∈ S , we have d(z , 0) ≤ d(g · z , 0), i.e. if z ∈ Ext(S).

Algorithm

Input: z ∈ B.Let z ′ = z . If d(g · z ′, 0) < d(z ′, 0) for some g ∈ S , thenset z ′ = g · z ′ and repeat.Output: z ′ ∈ B, S-reduced and δ ∈ 〈S〉 s.t. z ′ = δ · z .

AlgorithmsExamples

If a point z has a trivial stabilizer, then the orbit map γ 7→ γ · z isa bijection. This leads to

Definition

Let Γ be a Kleinian group, S ⊂ Γ and z ∈ B. An element γ ∈ Γis (S , z)-reduced if γ · z is S-reduced, i.e. if γ · z ∈ Ext(S).Thereduced element computed by the previous algorithm iswritten RedS(γ; z) and simply RedS(γ) = RedS(γ; 0).

Proposition

Suppose that Ext(S) is a fundamental domain for 〈S〉. Forall z ∈ Ext(S), we have γ = 1 if and only if γ ∈ 〈S〉.

AlgorithmsExamples

Definition

Proposition

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Definition

Proposition

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Definition

Proposition

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Algorithm

Enumerate some elements of Γ in S

(S , 0)-reduce the elements of S

For every z ∈ I(γ) s.t. γ · z /∈ Ext(S), add RedS(γ; z) to S

until Vol(Ext(S)) = Covol(Γ)

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Algorithm

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Algorithm

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Algorithm

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Bianchi groupsCocompact groups

Proposition

Let Γ = PSL2(Z[ 1+√−1632 ]). Then the Bianchi group Γ has

covolume Covol(Γ) ≈ 57.435648, and Γ admits a presentationwith 10 generators, 7 elliptic relations and 10 other relations.

The fundamental polyhedron that was computed has 111 facesand 306 edges, and the maximum absolute reduced norm of theelements that bound this exterior domain is 4536. In the lattice, 70millions of vectors were enumerated, and 8500 of them had norm 1.

AlgorithmsExamples

Proposition

Let F = Q( 3√

11) with discriminant −3267 and class number 2,

α = 3√

11, B =(−2,−4α2−α−2

), O a maximal order in B

and Γ = O×1 /± 1. The quaternion algebra B has discriminant p2

where N(p2) = 2. Then the group Γ hascovolume Covol(Γ) ≈ 206.391784, and Γ admits a presentationwith 17 generators, 11 elliptic relations and 21 other relations.

The fundamental polyhedron that was computed has 647 facesand 1877 edges, and the maximum absolute reduced norm of theelements that bound this exterior domain is 5802. In the lattice, 80millions of vectors were enumerated, and 300 of them had norm 1.

AlgorithmsExamples

computing fundamental domains for arithmetic kleinian...

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