Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Minimal co-volume hyperbolic lattices
Solution of Siegel’s problem in three dimensions
Gaven J. Martin
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
A lattice is a discrete subgroup Γ of isometries of hyperbolic 3-space H3
with finite co-volume. Thus the orbit space
O = H3 /Γ
is a hyperbolic orbifold, or manifold if Γ is torsion free (no elements of
finite order), of finite volume.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
A lattice is a discrete subgroup Γ of isometries of hyperbolic 3-space H3
with finite co-volume. Thus the orbit space
O = H3 /Γ
is a hyperbolic orbifold, or manifold if Γ is torsion free (no elements of
finite order), of finite volume.
An isometry Φ : H3 → H3 is uniquely determined by its boundary values
on ∂H3 ≈ Ĉ (Poincaré extension) which determine a Möbius
transformation and the maps
Isom+ (H3 ) 3 Φ ↔ Φ|Ĉ =
az + b
a
↔
c
cz + d
b
d
∈ PSL(2, C)
is an isomorphism of Lie groups.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
A lattice is a discrete subgroup Γ of isometries of hyperbolic 3-space H3
with finite co-volume. Thus the orbit space
O = H3 /Γ
is a hyperbolic orbifold, or manifold if Γ is torsion free (no elements of
finite order), of finite volume.
An isometry Φ : H3 → H3 is uniquely determined by its boundary values
on ∂H3 ≈ Ĉ (Poincaré extension) which determine a Möbius
transformation and the maps
Isom+ (H3 ) 3 Φ ↔ Φ|Ĉ =
az + b
a
↔
c
cz + d
b
d
∈ PSL(2, C)
is an isomorphism of Lie groups.
A Kleinian group is a discrete subgroup of Isom+ (H3 ) ≈ PSL(2, C)
which is not virtually abelian.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
In 1945 Siegel posed the problem of identifying the smallest co-volume
hyperbolic lattices in n dimensions
µ(n) = inf
Γ
Hn /Γ
Here Γ ⊂ Isom+ (Hn ) is a discrete subgroup.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
In 1945 Siegel posed the problem of identifying the smallest co-volume
hyperbolic lattices in n dimensions
µ(n) = inf
Γ
Hn /Γ
Here Γ ⊂ Isom+ (Hn ) is a discrete subgroup.
For Euclidean lattices (Bieberbach or Crystallographic groups) such an
infimum is obviously 0.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
In 1945 Siegel posed the problem of identifying the smallest co-volume
hyperbolic lattices in n dimensions
µ(n) = inf
Γ
Hn /Γ
Here Γ ⊂ Isom+ (Hn ) is a discrete subgroup.
In two dimensions Siegel shows
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
In 1945 Siegel posed the problem of identifying the smallest co-volume
hyperbolic lattices in n dimensions
µ(n) = inf
Γ
Hn /Γ
Here Γ ⊂ Isom+ (Hn ) is a discrete subgroup.
In two dimensions Siegel shows
µ(2) =
π
21 ,
attained for the (2, 3, 7)-triangle group.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
In 1945 Siegel posed the problem of identifying the smallest co-volume
hyperbolic lattices in n dimensions
µ(n) = inf
Γ
Hn /Γ
Here Γ ⊂ Isom+ (Hn ) is a discrete subgroup.
In two dimensions Siegel shows
µ(2) =
π
21 ,
attained for the (2, 3, 7)-triangle group.
Poincaré-Klein refined to signature formula
X
(1 − 1/m)
A = 2π 2g − 2 + n +
G.J.Martin
Minimal co-volume hyperbolic lattices
.
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
In 1945 Siegel posed the problem of identifying the smallest co-volume
hyperbolic lattices in n dimensions
µ(n) = inf
Γ
Hn /Γ
Here Γ ⊂ Isom+ (Hn ) is a discrete subgroup.
In two dimensions Siegel shows
µ(2) =
π
21 ,
attained for the (2, 3, 7)-triangle group.
Poincaré-Klein refined to signature formula
X
(1 − 1/m)
A = 2π 2g − 2 + n +
suggests a connection with Hurwitz’ 84g − 84
theorem on the symmetries of Riemann surfaces.
(confirmed by McBeath in 1961)
G.J.Martin
Minimal co-volume hyperbolic lattices
.
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
Mostow rigidity (topology = geometry in n ≥ 3).
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
Mostow rigidity (topology = geometry in n ≥ 3).
Kazhdan and Margulis show µ(n) > 0 and is attained.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
Mostow rigidity (topology = geometry in n ≥ 3).
Kazhdan and Margulis show µ(n) > 0 and is attained.
Wang shows spectrum of volumes of lattices is discrete for n ≥ 4
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
Mostow rigidity (topology = geometry in n ≥ 3).
Kazhdan and Margulis show µ(n) > 0 and is attained.
Wang shows spectrum of volumes of lattices is discrete for n ≥ 4
Selberg’s lemma gives general existence of torsion free subgroups of
finite index : the most symmetric hyperbolic manifolds are quotients
of smallest co-volume lattices.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
Mostow rigidity (topology = geometry in n ≥ 3).
Kazhdan and Margulis show µ(n) > 0 and is attained.
Wang shows spectrum of volumes of lattices is discrete for n ≥ 4
Selberg’s lemma gives general existence of torsion free subgroups of
finite index : the most symmetric hyperbolic manifolds are quotients
of smallest co-volume lattices.
Jørgensen, Thurston show in three dimensions that the set of
volumes is of type ω ω .
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
Mostow rigidity (topology = geometry in n ≥ 3).
Kazhdan and Margulis show µ(n) > 0 and is attained.
Wang shows spectrum of volumes of lattices is discrete for n ≥ 4
Selberg’s lemma gives general existence of torsion free subgroups of
finite index : the most symmetric hyperbolic manifolds are quotients
of smallest co-volume lattices.
Jørgensen, Thurston show in three dimensions that the set of
volumes is of type ω ω .
√
Meyerhoff µ(3) > 0.00005 and identifies PGL(2, O( −3)) as
smallest co-volume noncompact lattice.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
Mostow rigidity (topology = geometry in n ≥ 3).
Kazhdan and Margulis show µ(n) > 0 and is attained.
Wang shows spectrum of volumes of lattices is discrete for n ≥ 4
Selberg’s lemma gives general existence of torsion free subgroups of
finite index : the most symmetric hyperbolic manifolds are quotients
of smallest co-volume lattices.
Jørgensen, Thurston show in three dimensions that the set of
volumes is of type ω ω .
√
Meyerhoff µ(3) > 0.00005 and identifies PGL(2, O( −3)) as
smallest co-volume noncompact lattice.
Borel gives co-volume formula for maximal arithmetic lattices in
three dimensions. Gives useful criteria to determine arithmetity.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
Mostow rigidity (topology = geometry in n ≥ 3).
Kazhdan and Margulis show µ(n) > 0 and is attained.
Wang shows spectrum of volumes of lattices is discrete for n ≥ 4
Selberg’s lemma gives general existence of torsion free subgroups of
finite index : the most symmetric hyperbolic manifolds are quotients
of smallest co-volume lattices.
Jørgensen, Thurston show in three dimensions that the set of
volumes is of type ω ω .
√
Meyerhoff µ(3) > 0.00005 and identifies PGL(2, O( −3)) as
smallest co-volume noncompact lattice.
Borel gives co-volume formula for maximal arithmetic lattices in
three dimensions. Gives useful criteria to determine arithmetity.
Chinburg and Friedman identify smallest arithmetic lattice
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
In a sequence of works, culminating in papers with Gehring and with
Marshall we proved
Theorem
µ(3) = 2753/2 2−7 π −6 ζK (2) ≈ 0.03905
Here ζK is the Dedekind zeta function of the field
K = Q(α) with α a complex root of
α4 + 6α3 + 12α2 + 9α + 1 = 0
The extremal is uniquely achieved in the Z2 -extension of
the 3-5-3 hyperbolic Coxeter group.
This is an arithmetic two generator group, generated by
elements of orders 2 and 5.
G.J.Martin
Minimal co-volume hyperbolic lattices
Fred Gehring Tim Marshall Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Definitions
The problem
Progress and related results
The main result
The next three smallest co-volume lattices contain groups generated by
two elements of finite orders 2 and 3 with low index.
2833/2 2−7 π −6 ζK (2) = 0.0408 . . ., K = Q(γ),
γ 4 + 5γ 3 + 7γ 2 + 3γ + 1 = 0, discriminant −283 with unramified
quaternion algebra. It also is a two-generator arithmetic Kleinian
group.
313/2 2−6 π −4 (NP3 − 1)ζK (2) = 0.0659 . . ., K = Q(γ),
γ 3 + 4γ 2 + 5γ + 3 = 0, discriminant −31 and ramified at the finite
place P3 . It contains (3,0)-(3,0) Dehn sugery on the Whitehead link
of index 8 and a group generated by elements of order 2 and 3 of
index 4.
443/2 2−6 π −4 (NP2 − 1)ζk (2) = 0.0661 . . ., K = Q(γ),
γ 3 + 4γ 2 + 4γ + 2 = 0 of discriminant −44 and ramified at the
finite place P2 . It also contains a group generated by elements of
order 2 and 3 of index 4.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Basically an arithmetic Kleinian group Γ is one algebraically isomorphic to
something commensurable with SL(m, Z) for some m. So there’s a
representation ρ : Γ → GL(m, C) so that
|ρ(Γ) ∩ SL(m, Z) : SL(m, Z)| + |ρ(Γ) ∩ SL(m, Z) : ρ(Γ)| < ∞
Borel gave a nice description of these groups in three dimensions. The
group will be 2 × 2 matrices defined over a finite field (likely to be of
degree m) of special type with structure conditions on the associated
quaternion algebra.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Basically an arithmetic Kleinian group Γ is one algebraically isomorphic to
something commensurable with SL(m, Z) for some m. So there’s a
representation ρ : Γ → GL(m, C) so that
|ρ(Γ) ∩ SL(m, Z) : SL(m, Z)| + |ρ(Γ) ∩ SL(m, Z) : ρ(Γ)| < ∞
Borel gave a nice description of these groups in three dimensions. The
group will be 2 × 2 matrices defined over a finite field (likely to be of
degree m) of special type with structure conditions on the associated
quaternion algebra.
In higher rank, all lattices in semi-simple Lie groups are arithmetic
(Margulis super-rigidity) and even in rank one only hyperbolic and
complex hyperbolic lattices need not be arithmetic.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Signature formula and Takeuchi’s identification of arithmetic triangle
groups shows smallest 9 lattices of H2 are arithmetic. Smallest
non-arithmetic lattice is the (2, 3, 13)-triangle group.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Signature formula and Takeuchi’s identification of arithmetic triangle
groups shows smallest 9 lattices of H2 are arithmetic. Smallest
non-arithmetic lattice is the (2, 3, 13)-triangle group.
Two smallest noncompact lattices are arithmetic - (2, 3, ∞) and
(2, 4, ∞). Smallest non-arithmetic lattice is (2, 5, ∞).
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Signature formula and Takeuchi’s identification of arithmetic triangle
groups shows smallest 9 lattices of H2 are arithmetic. Smallest
non-arithmetic lattice is the (2, 3, 13)-triangle group.
Two smallest noncompact lattices are arithmetic - (2, 3, ∞) and
(2, 4, ∞). Smallest non-arithmetic lattice is (2, 5, ∞).
For 3-manifolds we have the expected data (Maclachlan-Reid)
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Signature formula and Takeuchi’s identification of arithmetic triangle
groups shows smallest 9 lattices of H2 are arithmetic. Smallest
non-arithmetic lattice is the (2, 3, 13)-triangle group.
Two smallest noncompact lattices are arithmetic - (2, 3, ∞) and
(2, 4, ∞). Smallest non-arithmetic lattice is (2, 5, ∞).
For 3-manifolds we have the expected data (Maclachlan-Reid)
Five smallest closed manifolds arithmetic but only 20 of first 50
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Signature formula and Takeuchi’s identification of arithmetic triangle
groups shows smallest 9 lattices of H2 are arithmetic. Smallest
non-arithmetic lattice is the (2, 3, 13)-triangle group.
Two smallest noncompact lattices are arithmetic - (2, 3, ∞) and
(2, 4, ∞). Smallest non-arithmetic lattice is (2, 5, ∞).
For 3-manifolds we have the expected data (Maclachlan-Reid)
Five smallest closed manifolds arithmetic but only 20 of first 50
Smallest noncompact manifold arithmetic but only 2 of first 50
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Signature formula and Takeuchi’s identification of arithmetic triangle
groups shows smallest 9 lattices of H2 are arithmetic. Smallest
non-arithmetic lattice is the (2, 3, 13)-triangle group.
Two smallest noncompact lattices are arithmetic - (2, 3, ∞) and
(2, 4, ∞). Smallest non-arithmetic lattice is (2, 5, ∞).
For 3-manifolds we have the expected data (Maclachlan-Reid)
Five smallest closed manifolds arithmetic but only 20 of first 50
Smallest noncompact manifold arithmetic but only 2 of first 50
In general not a lot of data - in joint work with Gehring, Maclachlan and
Reid we expect the dozen smallest lattices arithmetic,
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Signature formula and Takeuchi’s identification of arithmetic triangle
groups shows smallest 9 lattices of H2 are arithmetic. Smallest
non-arithmetic lattice is the (2, 3, 13)-triangle group.
Two smallest noncompact lattices are arithmetic - (2, 3, ∞) and
(2, 4, ∞). Smallest non-arithmetic lattice is (2, 5, ∞).
For 3-manifolds we have the expected data (Maclachlan-Reid)
Five smallest closed manifolds arithmetic but only 20 of first 50
Smallest noncompact manifold arithmetic but only 2 of first 50
In general not a lot of data - in joint work with Gehring, Maclachlan and
Reid we expect the dozen smallest lattices arithmetic,
.0390, .0408, .0659, .0661, .0717, .0785, .0845, .0933, .1028, .1268, .1274, .1374
Smallest non-arithmetic commensurable with a tetrahedral group.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Signature formula and Takeuchi’s identification of arithmetic triangle
groups shows smallest 9 lattices of H2 are arithmetic. Smallest
non-arithmetic lattice is the (2, 3, 13)-triangle group.
Two smallest noncompact lattices are arithmetic - (2, 3, ∞) and
(2, 4, ∞). Smallest non-arithmetic lattice is (2, 5, ∞).
For 3-manifolds we have the expected data (Maclachlan-Reid)
Five smallest closed manifolds arithmetic but only 20 of first 50
Smallest noncompact manifold arithmetic but only 2 of first 50
In general not a lot of data - in joint work with Gehring, Maclachlan and
Reid we expect the dozen smallest lattices arithmetic,
.0390, .0408, .0659, .0661, .0717, .0785, .0845, .0933, .1028, .1268, .1274, .1374
Smallest non-arithmetic commensurable with a tetrahedral group.
Prevalence of two generator arithmetic groups
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
simple and non-simple elements
spherical points
Any nonparabolic f element of a Kleinian group has two fixed points on
Ĉ and setwise fixes the closed hyperbolic line connecting these points the axis. Up to conjugacy these points may be {0, ∞} and the map is
conjugate to
α
0 z 7→ α2 z ∼
0 1/α
Ç
The map translates along the axis distance
τ (f ) = 2 log |α|,
C
(translation length) and rotates about the axis
an axis in H3
θ(f ) = 2 arg α,
the holonomy.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
simple and non-simple elements
spherical points
An axis α in Γ is simple if for each g ∈ Γ
g (α) = α,
or
G.J.Martin
g (α) ∩ α = ∅
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
simple and non-simple elements
spherical points
An axis α in Γ is simple if for each g ∈ Γ
g (α) = α,
or
g (α) ∩ α = ∅
The axis with shortest translation length is always simple (cut and paste
near the crossing).
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
simple and non-simple elements
spherical points
An axis α in Γ is simple if for each g ∈ Γ
g (α) = α,
or
g (α) ∩ α = ∅
The axis with shortest translation length is always simple (cut and paste
near the crossing). Discreteness shows if an axis α is simple, then
r=
n
o
1
inf ρH (α, g (α)) : g ∈ Γ, g (α) 6= α > 0
2
so α lies in a precisely invariant solid hyperbolic cylinder C(α, r ).
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
simple and non-simple elements
spherical points
An axis α in Γ is simple if for each g ∈ Γ
g (α) = α,
or
g (α) ∩ α = ∅
The axis with shortest translation length is always simple (cut and paste
near the crossing). Discreteness shows if an axis α is simple, then
r=
n
o
1
inf ρH (α, g (α)) : g ∈ Γ, g (α) 6= α > 0
2
so α lies in a precisely invariant solid hyperbolic cylinder C(α, r ). Clearly
volH (H3 /Γ) ≥ volH (C/Γα ) =
πτ
sinh2 (r )
p
as the (set) stabiliser Γα of α has a particularly simple structure - at worst
its Z × Zp o Z2 .
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
simple and non-simple elements
spherical points
An axis α in Γ is simple if for each g ∈ Γ
g (α) = α,
or
g (α) ∩ α = ∅
The axis with shortest translation length is always simple (cut and paste
near the crossing). Discreteness shows if an axis α is simple, then
r=
n
o
1
inf ρH (α, g (α)) : g ∈ Γ, g (α) 6= α > 0
2
so α lies in a precisely invariant solid hyperbolic cylinder C(α, r ). Clearly
volH (H3 /Γ) ≥ volH (C/Γα ) =
πτ
sinh2 (r )
p
as the (set) stabiliser Γα of α has a particularly simple structure - at worst
its Z × Zp o Z2 . Must bound r , τ from below and p from above.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
simple and non-simple elements
spherical points
An axis α in Γ is simple if for each g ∈ Γ
g (α) = α,
or
g (α) ∩ α = ∅
The axis with shortest translation length is always simple (cut and paste
near the crossing). Discreteness shows if an axis α is simple, then
r=
n
o
1
inf ρH (α, g (α)) : g ∈ Γ, g (α) 6= α > 0
2
so α lies in a precisely invariant solid hyperbolic cylinder C(α, r ). Clearly
volH (H3 /Γ) ≥ volH (C/Γα ) =
πτ
sinh2 (r )
p
as the (set) stabiliser Γα of α has a particularly simple structure - at worst
its Z × Zp o Z2 . Must bound r , τ from below and p from above.
Never sharp !
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
simple and non-simple elements
spherical points
If α is the axis of an elliptic f and g (α) ∩ α = x0 ∈ H3 , then f and
g ◦ f ◦ g −1 share a fixed point x0 and hf , g ◦ f ◦ g −1 i is conjugate to a
group of rotations - spherical triangle group.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
simple and non-simple elements
spherical points
If α is the axis of an elliptic f and g (α) ∩ α = x0 ∈ H3 , then f and
g ◦ f ◦ g −1 share a fixed point x0 and hf , g ◦ f ◦ g −1 i is conjugate to a
group of rotations - spherical triangle group.
These are classified as A4 , A5 and S4 and the dihedral groups Dn , n ≥ 2.
We call such a point x0 a spherical fixed point.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
simple and non-simple elements
spherical points
If α is the axis of an elliptic f and g (α) ∩ α = x0 ∈ H3 , then f and
g ◦ f ◦ g −1 share a fixed point x0 and hf , g ◦ f ◦ g −1 i is conjugate to a
group of rotations - spherical triangle group.
These are classified as A4 , A5 and S4 and the dihedral groups Dn , n ≥ 2.
We call such a point x0 a spherical fixed point.
Again discreteness gives
r=
n
o
1
inf ρH (x0 , g (x0 )) : g ∈ Γ \ Γx0 > 0
2
Thus BH (x0 , r ) is a precisely invariant ball and
volH (H3 /Γ) ≥ volH (B/Γx0 ) = π(sinh2 (2r ) − 2r )/|Γx0 |
so again we seek to bound r
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
simple and non-simple elements
spherical points
If α is the axis of an elliptic f and g (α) ∩ α = x0 ∈ H3 , then f and
g ◦ f ◦ g −1 share a fixed point x0 and hf , g ◦ f ◦ g −1 i is conjugate to a
group of rotations - spherical triangle group.
These are classified as A4 , A5 and S4 and the dihedral groups Dn , n ≥ 2.
We call such a point x0 a spherical fixed point.
Again discreteness gives
r=
n
o
1
inf ρH (x0 , g (x0 )) : g ∈ Γ \ Γx0 > 0
2
Thus BH (x0 , r ) is a precisely invariant ball and
volH (H3 /Γ) ≥ volH (B/Γx0 ) = π(sinh2 (2r ) − 2r )/|Γx0 |
so again we seek to bound r
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
simple and non-simple elements
spherical points
If α is the axis of an elliptic f and g (α) ∩ α = x0 ∈ H3 , then f and
g ◦ f ◦ g −1 share a fixed point x0 and hf , g ◦ f ◦ g −1 i is conjugate to a
group of rotations - spherical triangle group.
These are classified as A4 , A5 and S4 and the dihedral groups Dn , n ≥ 2.
We call such a point x0 a spherical fixed point.
Again discreteness gives
r=
n
o
1
inf ρH (x0 , g (x0 )) : g ∈ Γ \ Γx0 > 0
2
Thus BH (x0 , r ) is a precisely invariant ball and
volH (H3 /Γ) ≥ volH (B/Γx0 ) = π(sinh2 (2r ) − 2r )/|Γx0 |
so again we seek to bound r
Never sharp !
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
The question of bounding the quantities in the estimates above reduce to
questions about two generator discrete groups.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
The question of bounding the quantities in the estimates above reduce to
questions about two generator discrete groups.
Let Γ be a Kleinian group. The trace of f ∈ Γ is tr (f ) = ±tr (Af ) where
Af ∈ PSL(2, C) represents f . Then for f , g ∈ Isom+ (H)
β(f ) = tr 2 (f ) − 4, β(g ) = tr 2 (g ) − 4 and γ(f , g ) = tr [f , g ] − 2
These parameters determine hf , g i uniquely up to conjugacy and encode
other geometric quantities.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
γ(f , g ) = 0 if and only if f and g share a common fixed point in Ĉ
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
γ(f , g ) = 0 if and only if f and g share a common fixed point in Ĉ
f and g are elliptic or loxodromic with translation lengths τf and τg
and holonomies θf and θg , then
τf + iθf
2
β(f ) = 4 sinh
2
τ
+
iθg
g
2
β(g ) = 4 sinh
2
β(f )β(g )
γ(f , g ) =
sinh2 (δ + iφ)
4
where δ + iφ is the complex distance between the two axes.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
View the space of all two-generator Kleinian groups modulo conjugacy as
a subset of C3 via the map
hf , g i → (γ(f , g ), β(f ), β(g ))
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
View the space of all two-generator Kleinian groups modulo conjugacy as
a subset of C3 via the map
hf , g i → (γ(f , g ), β(f ), β(g ))
The fundamental problem is to find the points in C3 which correspond to
discrete groups - or at least give a good description of this space.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
View the space of all two-generator Kleinian groups modulo conjugacy as
a subset of C3 via the map
hf , g i → (γ(f , g ), β(f ), β(g ))
The fundamental problem is to find the points in C3 which correspond to
discrete groups - or at least give a good description of this space.
This space has very complicated structure, but those points
corresponding to two-generator lattices will be isolated.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
View the space of all two-generator Kleinian groups modulo conjugacy as
a subset of C3 via the map
hf , g i → (γ(f , g ), β(f ), β(g ))
The fundamental problem is to find the points in C3 which correspond to
discrete groups - or at least give a good description of this space.
This space has very complicated structure, but those points
corresponding to two-generator lattices will be isolated.
Lemma (Projection)
(γ(f , g ), β(f ), β(g )) discrete ⇒ (γ, β, −4) discrete
This map is a contraction on complex distances. Away from a finite set
of exceptional parameters this preserves “non-elementary”.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Let ha, bi be the free group on the two letters a and b. Say w ∈ ha, bi is
a good word if w can be written as
w = b s1 ar1 b s2 ar2 . . . b sm−1 arm−1 b sm
where s1 ∈ {±1}, sj = (−1)j+1 s1 and rj 6= 0
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Let ha, bi be the free group on the two letters a and b. Say w ∈ ha, bi is
a good word if w can be written as
w = b s1 ar1 b s2 ar2 . . . b sm−1 arm−1 b sm
where s1 ∈ {±1}, sj = (−1)j+1 s1 and rj 6= 0. Good words start with b
and end in b ±1 depending on whether m is even or odd - the exponents
of b alternate in sign
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Let ha, bi be the free group on the two letters a and b. Say w ∈ ha, bi is
a good word if w can be written as
w = b s1 ar1 b s2 ar2 . . . b sm−1 arm−1 b sm
where s1 ∈ {±1}, sj = (−1)j+1 s1 and rj 6= 0. Good words start with b
and end in b ±1 depending on whether m is even or odd - the exponents
of b alternate in sign.
Next we introduce a key tool used in our study of parameter spaces of
discrete groups.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Let ha, bi be the free group on the two letters a and b. Say w ∈ ha, bi is
a good word if w can be written as
w = b s1 ar1 b s2 ar2 . . . b sm−1 arm−1 b sm
where s1 ∈ {±1}, sj = (−1)j+1 s1 and rj 6= 0. Good words start with b
and end in b ±1 depending on whether m is even or odd - the exponents
of b alternate in sign.
Next we introduce a key tool used in our study of parameter spaces of
discrete groups.
Theorem
Let a, b ∈ PSL(2, C) and w = w (a, b) ∈ ha, bi be a good word. Set
β = β(f ) and γ = γ(f , g ). Then there is a monic polynomial with
integer coefficients such that
γ(f , w (f , g )) = pw (γ, β).
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
There are three things to note.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
There are three things to note.
◦ If b 2 = 1 : alternating sign is redundant and every w (a, b) is good.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
There are three things to note.
◦ If b 2 = 1 : alternating sign is redundant and every w (a, b) is good.
◦ A semigroup operation on good words: If w1 = w1 (a, b) and
w2 = w2 (a, b) are good , then so is
w1 ∗ w2 = w1 (a, w2 (a, b))
For example
(bab −1 ab) ∗ (bab −1 )
= bab −1 a(bab −1 )−1 abab −1
= bab −1 aba−1 b −1 abab −1
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
There are three things to note.
◦ If b 2 = 1 : alternating sign is redundant and every w (a, b) is good.
◦ A semigroup operation on good words: If w1 = w1 (a, b) and
w2 = w2 (a, b) are good , then so is
w1 ∗ w2 = w1 (a, w2 (a, b))
For example
(bab −1 ab) ∗ (bab −1 )
= bab −1 a(bab −1 )−1 abab −1
= bab −1 aba−1 b −1 abab −1
◦
pw1 ∗w2 (γ, β) = pw1 (pw2 (γ, β), β)
corresponding to polynomial composition in the first slot.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
There are three things to note.
◦ If b 2 = 1 : alternating sign is redundant and every w (a, b) is good.
◦ A semigroup operation on good words: If w1 = w1 (a, b) and
w2 = w2 (a, b) are good , then so is
w1 ∗ w2 = w1 (a, w2 (a, b))
For example
(bab −1 ab) ∗ (bab −1 )
= bab −1 a(bab −1 )−1 abab −1
= bab −1 aba−1 b −1 abab −1
◦
pw1 ∗w2 (γ, β) = pw1 (pw2 (γ, β), β)
corresponding to polynomial composition in the first slot.
Generalised Chebychev polynomials ??
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Notice the obvious facts
◦ hf , g i Kleinian, implies hf , w (f , g )i discrete.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Notice the obvious facts
◦ hf , g i Kleinian, implies hf , w (f , g )i discrete.
◦ for any word w = w (f , g ) and m, n ∈ Z, γ(f , f m wf n ) = γ(f , w ) - the
requirement the word start and end in a nontrivial power of b is simply to
avoid obvious redundancy.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Notice the obvious facts
◦ hf , g i Kleinian, implies hf , w (f , g )i discrete.
◦ for any word w = w (f , g ) and m, n ∈ Z, γ(f , f m wf n ) = γ(f , w ) - the
requirement the word start and end in a nontrivial power of b is simply to
avoid obvious redundancy.
Two simple examples of word polynomials and how they generate
inequalities. (later computer assisted searches amount in large part to
mechanising these arguments).
We fix β and write z for γ(f , g ) and use this as a variable. Then
w = bab −1
w = bab
−1
ab
w = bab −1 a−1 b
pw (z) = z(z − β)
pw (z) = z(1 + β − z)2
pw (z) = z(1 − 2β + 2z − βz + z 2 )
We indicate how these words are used to describe parts of the parameter
space for two-generator Kleinian groups.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Notice the obvious facts
◦ hf , g i Kleinian, implies hf , w (f , g )i discrete.
◦ for any word w = w (f , g ) and m, n ∈ Z, γ(f , f m wf n ) = γ(f , w ) - the
requirement the word start and end in a nontrivial power of b is simply to
avoid obvious redundancy.
Two simple examples of word polynomials and how they generate
inequalities. (later computer assisted searches amount in large part to
mechanising these arguments).
We fix β and write z for γ(f , g ) and use this as a variable. Then
w = bab −1
w = bab
−1
ab
w = bab −1 a−1 b
pw (z) = z(z − β)
pw (z) = z(1 + β − z)2
pw (z) = z(1 − 2β + 2z − βz + z 2 )
We indicate how these words are used to describe parts of the parameter
space for two-generator Kleinian groups. We take for granted the well
known fact that the space of discrete non-elementary groups is closed
(a general fact concerning groups of isometries of negative curvature).
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Let’s recover Jørgensen’s inequality.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Let’s recover Jørgensen’s inequality. Consider
J = inf{ |γ| + |β| : (γ, β, β 0 ) parameters of a Kleinian group}.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Let’s recover Jørgensen’s inequality. Consider
J = inf{ |γ| + |β| : (γ, β, β 0 ) parameters of a Kleinian group}.
The minimum is attained by Γ = hf , g i.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Let’s recover Jørgensen’s inequality. Consider
J = inf{ |γ| + |β| : (γ, β, β 0 ) parameters of a Kleinian group}.
The minimum is attained by Γ = hf , g i. Consider Γ0 = hf , gfg −1 i.
By minimality,
|γ| + |β|
≤
|γ(γ − β)| + |β|
1
≤
|γ − β| ≤ |γ| + |β|
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Let’s recover Jørgensen’s inequality. Consider
J = inf{ |γ| + |β| : (γ, β, β 0 ) parameters of a Kleinian group}.
The minimum is attained by Γ = hf , g i. Consider Γ0 = hf , gfg −1 i.
By minimality,
|γ| + |β|
≤
|γ(γ − β)| + |β|
1
≤
|γ − β| ≤ |γ| + |β|
We need to analyse the zero locus of the polynomial (in this case the
variety γ = β)
Theorem (Jørgensen’s inequality)
Let G = hf , g i be a Kleinian group. Then
|γ(f , g )| + |β(f )| ≥ 1
This inequality holds with equality for (2, 3, p)-triangle groups, p ≥ 7.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Jørgensen’s inequality implies that geodesics cannot be too short without
making a specific volume contribution.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Jørgensen’s inequality implies that geodesics cannot be too short without
making a specific volume contribution. If αf is the axis of f and α̃f its
nearest translate (by g ) at complex distance δ + iθ, we compute
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Jørgensen’s inequality implies that geodesics cannot be too short without
making a specific volume contribution. If αf is the axis of f and α̃f its
nearest translate (by g ) at complex distance δ + iθ, we compute
| sinh2 (δ + iθ)|
≥
sinh2 (2r ) ≥
πτ sinh2 (r ) ≥
γ 1 − |β|
≥
β
|β|
√
1 − 2|β|
3 − 8πτ
≈
|β|
4πτ
√
3 − 8πτ
≈ 0.108 . . .
16
≈ follows from a diaphantine analysis on the holonomy (replace f by f n )
and
n τ
n θ 4πτ
min β(f n ) = min 4 sinh2
+i
≈ √
n
n
2
2
3
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
As far as our search for Kleinian group parameters in C3 is concerned,
Jørgensen’s inequality tell us that the region
{(γ, β, β 0 ) : |γ| + |β| < 1 or |γ| + |β 0 | < 1}
contains no parameters for Kleinian groups - eliminating a quantitative
neighbourhood of (0, 0, 0) the identity group.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
As far as our search for Kleinian group parameters in C3 is concerned,
Jørgensen’s inequality tell us that the region
{(γ, β, β 0 ) : |γ| + |β| < 1 or |γ| + |β 0 | < 1}
contains no parameters for Kleinian groups - eliminating a quantitative
neighbourhood of (0, 0, 0) the identity group.
A point to observe here is that in more general situations we must
examine and eliminate, for some geometric reason, the zero locus of pw
(arithmeticity !)
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
As far as our search for Kleinian group parameters in C3 is concerned,
Jørgensen’s inequality tell us that the region
{(γ, β, β 0 ) : |γ| + |β| < 1 or |γ| + |β 0 | < 1}
contains no parameters for Kleinian groups - eliminating a quantitative
neighbourhood of (0, 0, 0) the identity group.
A point to observe here is that in more general situations we must
examine and eliminate, for some geometric reason, the zero locus of pw
(arithmeticity !)
Next, if we minimize |γ| + |1 + β| and use the second polynomial
z(1 + β − z)2 we see that at the minimum
|γ| + |1 + β| ≤
1
|γ(1 + β − γ)2 | + |1 + β|
≤ |1 + β − γ|
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Lemma
Let hf , g i be a Kleinian group. Then
|γ| + |1 + β| ≥ 1
unless γ = 1 + β (in which case fg or fg −1 is elliptic of order 3).
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Lemma
Let hf , g i be a Kleinian group. Then
|γ| + |1 + β| ≥ 1
unless γ = 1 + β (in which case fg or fg −1 is elliptic of order 3).
The zero locus we consider is {γ = 1 + β}; groups are Nielsen equivalent
to those generated by elliptics of order 2 and 3.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Lemma
Let hf , g i be a Kleinian group. Then
|γ| + |1 + β| ≥ 1
unless γ = 1 + β (in which case fg or fg −1 is elliptic of order 3).
The zero locus we consider is {γ = 1 + β}; groups are Nielsen equivalent
to those generated by elliptics of order 2 and 3.
As a consequence, if f has order 6, then β = −1 and we have
|γ(f , g )| ≥ 1 analogous to Shimitzu-Leutbecher. This quickly gives us
another useful result
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Theorem
Let Γ be a Kleinian group with an elliptic of order p ≥ 6, then
√
vol(H3 /Γ) ≥ vol(H3 /PGL(2, O( −3)) = 0.0846 . . .
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Theorem
Let Γ be a Kleinian group with an elliptic of order p ≥ 6, then
√
vol(H3 /Γ) ≥ vol(H3 /PGL(2, O( −3)) = 0.0846 . . .
It is relatively easy to get a bound bigger than this for torsion free lattices
(the manifold case). However, recently Gabai, Meyerhoff and Milley
identified the Weeks manifold - (5, 2) and (5, 1) Dehn surgery on the
Whitehead link (vol = 0.9427 . . .) - as having minimal volume.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Theorem
Let Γ be a Kleinian group with an elliptic of order p ≥ 6, then
√
vol(H3 /Γ) ≥ vol(H3 /PGL(2, O( −3)) = 0.0846 . . .
It is relatively easy to get a bound bigger than this for torsion free lattices
(the manifold case). However, recently Gabai, Meyerhoff and Milley
identified the Weeks manifold - (5, 2) and (5, 1) Dehn surgery on the
Whitehead link (vol = 0.9427 . . .) - as having minimal volume.
So we can focus on lattices which are not torsion free and for which all
the elements of finite order lie are 2, 3, 4 or 5.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
We want to consider groups generated by two elliptic elements p and q.
In this case β(f ) = −4 sin2 (π/p) and β(g ) = −4 sin2 (π/q) and the free
parameter is γ = γ(f , g ) ∈ C.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
We want to consider groups generated by two elliptic elements p and q.
In this case β(f ) = −4 sin2 (π/p) and β(g ) = −4 sin2 (π/q) and the free
parameter is γ = γ(f , g ) ∈ C.
f ∼
cos(π/p)
− sin(π/p)
sin(π/p) ,
cos(π/p)
g∼
cos(π/q)
− sin(π/q)/ω
ω sin(π/q) cos(π/q)
with |ω| ≤ 1 and
γ = −4 sin2 (π/p) sin2 (π/q)(ω − 1/ω)2
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
We want to consider groups generated by two elliptic elements p and q.
In this case β(f ) = −4 sin2 (π/p) and β(g ) = −4 sin2 (π/q) and the free
parameter is γ = γ(f , g ) ∈ C.
f ∼
cos(π/p)
− sin(π/p)
sin(π/p) ,
cos(π/p)
g∼
cos(π/q)
− sin(π/q)/ω
ω sin(π/q) cos(π/q)
with |ω| ≤ 1 and
γ = −4 sin2 (π/p) sin2 (π/q)(ω − 1/ω)2
Isometric circles of f are |z ± cot(π/p)| = 1/ sin(π/p)
and g are |z ± ω cot(π/p)| = |ω|/ sin(π/p) and these
are disjoint as soon as |γ| > 4.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
We want to consider groups generated by two elliptic elements p and q.
In this case β(f ) = −4 sin2 (π/p) and β(g ) = −4 sin2 (π/q) and the free
parameter is γ = γ(f , g ) ∈ C.
f ∼
cos(π/p)
− sin(π/p)
sin(π/p) ,
cos(π/p)
g∼
cos(π/q)
− sin(π/q)/ω
ω sin(π/q) cos(π/q)
with |ω| ≤ 1 and
γ = −4 sin2 (π/p) sin2 (π/q)(ω − 1/ω)2
Isometric circles of f are |z ± cot(π/p)| = 1/ sin(π/p)
and g are |z ± ω cot(π/p)| = |ω|/ sin(π/p) and these
are disjoint as soon as |γ| > 4.
Thus hf , g i is free on generators and not a lattice.
Also |γ| < 1 − 4 sin2 (π/p) gives hf , g i not discrete.
G.J.Martin
Minimal co-volume hyperbolic lattices
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
The (2, 3)-commutator plane.
COMMUTATOR PARAMETER FOR n= 3, m= 2
γ=
1
1
3
sinh2 (δ + iθ)
0.5
-4
-3
-2
-1
0
1
-0.5
-1
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Theorem (Gehring Maclachlan Martin Reid - after Borel)
Γ generated by elliptics of order p and q with γ = γ(f , g ). Γ is a discrete
subgroup of an arithmetic group if and only if γ is the root of a monic
polynomial with exactly one complex conjugate pair of roots and all the
real roots lie in the interval [−4 sin2 (π/p) sin2 (π/q), 0].
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Theorem (Gehring Maclachlan Martin Reid - after Borel)
Γ generated by elliptics of order p and q with γ = γ(f , g ). Γ is a discrete
subgroup of an arithmetic group if and only if γ is the root of a monic
polynomial with exactly one complex conjugate pair of roots and all the
real roots lie in the interval [−4 sin2 (π/p) sin2 (π/q), 0].
Theorem (Maclachlan Martin)
There are only finitely many arithmetic Kleinian groups generated by two
elements of orders p and q, 2 ≤ p, q ≤ ∞. In particular:
only finitely many arithmetic generalised triangle groups.
This is contrary to a conjecture of Hilden and Montesinos.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Theorem (Gehring Maclachlan Martin Reid - after Borel)
Γ generated by elliptics of order p and q with γ = γ(f , g ). Γ is a discrete
subgroup of an arithmetic group if and only if γ is the root of a monic
polynomial with exactly one complex conjugate pair of roots and all the
real roots lie in the interval [−4 sin2 (π/p) sin2 (π/q), 0].
Theorem (Maclachlan Martin)
There are only finitely many arithmetic Kleinian groups generated by two
elements of orders p and q, 2 ≤ p, q ≤ ∞. In particular:
only finitely many arithmetic generalised triangle groups.
This is contrary to a conjecture of Hilden and Montesinos.
What are they ? Serious computational number theory, using work of
Stark, Odlyzko, Diaz Y Diaz and Olivier on discriminant bounds for fields
as well as computer searches, then obtaining co-volume bounds and a
topological description.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Theorem
There are 41 non-uniform arithmetic lattices p, q-generated.
(∞, ∞) 4 two bridge knot & link complements
(2, ∞) 6 groups, (4, 6) 1 group
(3, ∞), (4, ∞), (2, 3), (2, 4), (2, 6), (3, 6) 3 groups each
(6, ∞) (3, 3), (3, 4), (4, 4), (6, 6) 2 groups each
There are 18 uniform arithmetic groups with p, q ≥ 6: (6, 6) 14, (8, 8) &
(10, 10), 1 each, (12, 12) - 2
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Theorem
There are 41 non-uniform arithmetic lattices p, q-generated.
(∞, ∞) 4 two bridge knot & link complements
(2, ∞) 6 groups, (4, 6) 1 group
(3, ∞), (4, ∞), (2, 3), (2, 4), (2, 6), (3, 6) 3 groups each
(6, ∞) (3, 3), (3, 4), (4, 4), (6, 6) 2 groups each
There are 18 uniform arithmetic groups with p, q ≥ 6: (6, 6) 14, (8, 8) &
(10, 10), 1 each, (12, 12) - 2
Corollary (a conjecture of Montesinos)
Let K be (p, 0), (q, 0) orbifold Dehn surgery on a two bridge knot or link
(p = q for knots) and K arithmetic. Then p, q ∈ {2, 3, 4, 6, 8, 10, 12}
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Here is the sort of data we pick up: Γ = hf , g i be discrete generated by
elliptics of orders p and q and let with δ + iθ be the complex distance
between the axes. Then
(p = 3, q = 2)
(p = 4, q = 2)
0.19707 + i0.78539 arithmetic
0.41572 + i0.59803 arithmetic
0.21084 + i0.33189 arithmetic
0.42698 + i0.44303 arithmetic
0.23371 + i0.49318 arithmetic
0.44068 + i0.78539 arithmetic
0.24486 + i0.67233 arithmetic
0.50495 + i0.67478 arithmetic
0.24809 + i0.40575 arithmetic
0.52254 + i0.34470 arithmetic
0.27407 + i0.61657 arithmetic
0.52979 + i0.24899 arithmetic
0.27465 + i0.78539 arithmetic
0.52979 + i0.53640 arithmetic
0.27702 + i0.56753 arithmetic
0.53063 arithmetic
0.27884 + i0.22832 arithmetic
0.53063 + i0.45227 arithmetic
δ > 0.28088
G.J.Martin
δ > 0.53264
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
(p = 3, q = 3)
(p = 2, q = 5)
0.39415 + i1.57079 arithmetic
0.4568 arithmetic
0.42168 + i0.66379 arithmetic
0.5306 arithmetic
0.46742 + i0.98637 arithmetic
0.6097 arithmetic
0.48973 + i1.34468 arithmetic
0.6268 arithmetic
0.49619 + i0.81150 arithmetic
0.6514 notarithmetic
0.54814 + i1.23135 arithmetic
0.6717 notarithmetic
0.54930 + i1.57079 arithmetic
0.6949 arithmetic
0.55404 + i1.13507 arithmetic
0.7195 arithmetic
0.55769 + i0.45665 arithmetic
0.7273 arithmetic
δ > 0.56177
G.J.Martin
δ > 0.73
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
From this data general position arguments on the placement of elliptic
axes
a3
b
yP
SF
Q
P
g
SG
g
g(Q)
and then an anlysis of what happens when the axes meet or coincide
yields data such as
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Let P and Q be spherical points in a Kleinian group (points stabilised by
a finite spherical triangle subgroup)
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Let P and Q be spherical points in a Kleinian group (points stabilised by
a finite spherical triangle subgroup)
P, Q tetrahedral points. If ρH (P, Q) < 1.026, then P and Q lie on
a common axis of order 3 and ρH (P, Q) = 0.64244 or 0.6931 or
ρH (P, Q) > 0.7209. Extremals are arithmetic
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Let P and Q be spherical points in a Kleinian group (points stabilised by
a finite spherical triangle subgroup)
P, Q tetrahedral points. If ρH (P, Q) < 1.026, then P and Q lie on
a common axis of order 3 and ρH (P, Q) = 0.64244 or 0.6931 or
ρH (P, Q) > 0.7209. Extremals are arithmetic
P, Q octahedral points. If ρH (P, Q) < 1.6140, then P and Q lie on
a common axis of order 4 and ρH (P, Q) = 1.0595 or 1.0612 or
1.1283 or ρH (P, Q) > 1.14. Extremals are arithmetic
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
Let P and Q be spherical points in a Kleinian group (points stabilised by
a finite spherical triangle subgroup)
P, Q tetrahedral points. If ρH (P, Q) < 1.026, then P and Q lie on
a common axis of order 3 and ρH (P, Q) = 0.64244 or 0.6931 or
ρH (P, Q) > 0.7209. Extremals are arithmetic
P, Q octahedral points. If ρH (P, Q) < 1.6140, then P and Q lie on
a common axis of order 4 and ρH (P, Q) = 1.0595 or 1.0612 or
1.1283 or ρH (P, Q) > 1.14. Extremals are arithmetic
P, Q icosahedral points. If ρH (P, Q) < 2.1225, then P and Q lie on
a common axis. If this axis has order 5, then ρH (P, Q) = 1.3825 or
1.6169 or ρH (P, Q) > 1.98968. If this axis has order 3, then
ρH (P, Q) ≥ 1.9028. Extremals are arithmetic
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
We are now left with the case that there are only Klein 4-groups, these
are wound up by discrete groups generated by two loxodromics whose
axes meet orthogonally.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
We are now left with the case that there are only Klein 4-groups, these
are wound up by discrete groups generated by two loxodromics whose
axes meet orthogonally. Then
Theorem
Let f and g be loxodromic generating a discrete group, the axes of f and
g perpendicular and τf and τg translation lengths. Then
!
√
3+1
max{τf , τg } ≥ λ⊥ = arccosh
= 0.831446 . . .
2
Equality holds for an arithmetic four fold cover of (4,0) & (2,0) Dehn
surgery on the 2 bridge link complement 622 of Rolfsen’s tables.
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
We are now left with the case that there are only Klein 4-groups, these
are wound up by discrete groups generated by two loxodromics whose
axes meet orthogonally. Then
Theorem
Let f and g be loxodromic generating a discrete group, the axes of f and
g perpendicular and τf and τg translation lengths. Then
!
√
3+1
max{τf , τg } ≥ λ⊥ = arccosh
= 0.831446 . . .
2
Equality holds for an arithmetic four fold cover of (4,0) & (2,0) Dehn
surgery on the 2 bridge link complement 622 of Rolfsen’s tables.
We expect that this represents the extreme case independently of the
angle at which the axes of loxodromics meet and that λ⊥ is the Margulis
number (for torsion free lattices)
G.J.Martin
Minimal co-volume hyperbolic lattices
Introduction
Arithmetic hyperbolic geometry
Geometry of discrete groups
Two generator groups
Parameters for two generator groups
Polynomial trace identities
Short geodesics
Eliminating large torsion
2-torsion and Klein 4 groups.
What next ?
Extend to identify next few lattices.
Identify all the arithmetic generalised triable groups with
2 ≤ p, q ≤ 5 (mostly done).
Use polynomial trace identities to reprove (log 3)/2 theorem.
Say a lot more about the commutator plane
boundary a circle ?
relations between different planes?
metrics ?
Conway notation ↔ pleating rays. Scattering ?
identify Margulis constant
G.J.Martin
Minimal co-volume hyperbolic lattices