# Scissors Congruence The Birth of Hyperbolic Volume by dfgh4bnmu

VIEWS: 12 PAGES: 70

• pg 1
```									Scissors Congruence: The Birth of Hyperbolic Volume
Gregory Leibon

Department of Mathematics
Dartmouth College

Scissors Congruence: The Birth of Hyperbolic Volume – p.1/70
The Ideal Tetrahedron

Here we see the oriented convex hull of four ideal points, an ideal tetrahedron.
8

8
8
8

Scissors Congruence: The Birth of Hyperbolic Volume – p.2/70
Ideal Tetrahedron

The boundary at inﬁnity is the Riemann sphere with hyperbolic isometries corresponding
to conformal mappings. Hence we label the points...

p
s

q              r

Scissors Congruence: The Birth of Hyperbolic Volume – p.3/70
Ideal Tetrahedron

..and compute the cross ratio z. This cross ration parameterizes these labeled oriented
ideal tetrahedra.

p   0
s

8

r
q

1       z=[p,q;r,s]

Scissors Congruence: The Birth of Hyperbolic Volume – p.4/70
Ideal Tetrahedron

It is easy to see that this cross ration depends really only on a choice of orientation and a
choice of a pair of opposite edges. Hence, the complex coordinate parameterize the
space of ideal oriented tetrahedra with a speciﬁed pair of opposite edges.

z
(z−1)/z
1/(1−z)
(z−1)/z
1/(1−z)

z

Scissors Congruence: The Birth of Hyperbolic Volume – p.5/70
Another Big Free Groups

Let
<C>
denote the free Abelian group generated
by all complex numbers, i.e. all ideal
tetrahedra.

Scissors Congruence: The Birth of Hyperbolic Volume – p.6/70
Key Relations

Two understand the need relations, take a pair of ideal tetrahedra and...

Scissors Congruence: The Birth of Hyperbolic Volume – p.7/70
Key Relations

and...

Scissors Congruence: The Birth of Hyperbolic Volume – p.8/70
Key Relations

and...

Scissors Congruence: The Birth of Hyperbolic Volume – p.9/70
Key Relations

and...

Scissors Congruence: The Birth of Hyperbolic Volume – p.10/70
Key Relations

and...

Scissors Congruence: The Birth of Hyperbolic Volume – p.11/70
Key Relations

and...

Scissors Congruence: The Birth of Hyperbolic Volume – p.12/70
Key Relations

glue them together.

Scissors Congruence: The Birth of Hyperbolic Volume – p.13/70
Key Relations

Now "ﬁrepole" this pair and...

Scissors Congruence: The Birth of Hyperbolic Volume – p.14/70
Key Relations

and...

Scissors Congruence: The Birth of Hyperbolic Volume – p.15/70
Key Relations

and...

Scissors Congruence: The Birth of Hyperbolic Volume – p.16/70
Key Relations

and...

Scissors Congruence: The Birth of Hyperbolic Volume – p.17/70
Key Relations

and...

Scissors Congruence: The Birth of Hyperbolic Volume – p.18/70
Key Relations

and...

Scissors Congruence: The Birth of Hyperbolic Volume – p.19/70
Key Relations

and...

Scissors Congruence: The Birth of Hyperbolic Volume – p.20/70
Key Relations

we can re-express this pair as three ideal tetrahedra. This is called a 2-3 move.

Scissors Congruence: The Birth of Hyperbolic Volume – p.21/70
Key Relations

In terms of the z coordinates we have

[z] + [w]

Scissors Congruence: The Birth of Hyperbolic Volume – p.22/70
Key Relations

equals

z − zw   w − zw
[zw] +        +
1 − zw   1 − zw

Scissors Congruence: The Birth of Hyperbolic Volume – p.23/70
The Relations

Let T the subgroup of < C > generated
by all elements in the form
z − zw   w − zw
[z] + [w] − [zw] −        −        ,
1 − zw   1 − zw
where z and w are complex numbers, to-
gether with all elements in the form [z] +
z
[¯].

Scissors Congruence: The Birth of Hyperbolic Volume – p.24/70
The Dupont and Sah Theorem

Wonderfully enough these are all the relations we need.   Theorem:
(Dupont, Sah)

∼ < C >,
Sis(H ) =    n
T

Scissors Congruence: The Birth of Hyperbolic Volume – p.25/70
The Proof

Recall Sis∞ (H n ) ∼ Sis(H n ). A key step in the proof is showing we can express a ﬁnite
=
tetrahedron using ideal tetrahedra.

Scissors Congruence: The Birth of Hyperbolic Volume – p.26/70
From Finite to Inﬁnite

Let us make a ﬁnite vertex inﬁnite. First extend an edge to inﬁnity.

8

Scissors Congruence: The Birth of Hyperbolic Volume – p.27/70
From Finite to Inﬁnite

Then form the red tetrahedra, with an ideal vertex.

+

+
−
8

Scissors Congruence: The Birth of Hyperbolic Volume – p.28/70
From Finite to Inﬁnite

and note...

+

+
−

Scissors Congruence: The Birth of Hyperbolic Volume – p.29/70
From Finite to Inﬁnite

and note...

+
+ −

Scissors Congruence: The Birth of Hyperbolic Volume – p.30/70
From Finite to Inﬁnite

and note...

+
−

Scissors Congruence: The Birth of Hyperbolic Volume – p.31/70
From Finite to Inﬁnite

and note...

−

+

Scissors Congruence: The Birth of Hyperbolic Volume – p.32/70
From Finite to Inﬁnite

and note...

−

+

Scissors Congruence: The Birth of Hyperbolic Volume – p.33/70
From Finite to Inﬁnite

Hence, we have expressed the ﬁnite tetrahedron using two ideal tetrahedra each with
only 3 ﬁnite vertices.

−

8
+
8

Scissors Congruence: The Birth of Hyperbolic Volume – p.34/70
From Finite to Inﬁnite

One can continue this till one is using
only ideal tetrahedra. The hard step is re-
moving the ﬁnal vertex. The best known
method to do this is due to Yana Mohanty
(2003).

Scissors Congruence: The Birth of Hyperbolic Volume – p.35/70
Getting a Grip on Volume

At this point, we see that understanding
hyperbolic volume can be reduced to un-
derstanding the volume of an ideal tetra-
hedron. To this it useful to take a close
look at the ideal tetrahedron’s angles.

Scissors Congruence: The Birth of Hyperbolic Volume – p.36/70
Ideal Tetrahedron’s Angles

Given any ideal polyhedron, at each ideal vertex we see this. The red sphere is a
horosphere.

8
A

B
C

Scissors Congruence: The Birth of Hyperbolic Volume – p.37/70
Euclidean Angles

Sending the ideal vertex to the point at inﬁnity in the upper-half space model, we ﬁnd that
the angles at an ideal vertex are Euclidean.

8
A                       C
B

Scissors Congruence: The Birth of Hyperbolic Volume – p.38/70
Ideal Tetrahedron’s Angles

We view our tetrahedron in the upper-half space model.

8

z
0                   1

Scissors Congruence: The Birth of Hyperbolic Volume – p.39/70
Ideal Tetrahedron’s Angles

Looking down from inﬁnity we see.

z

A+B+C= π                C

A                B
0                         1
Scissors Congruence: The Birth of Hyperbolic Volume – p.40/70
Ideal Tetrahedron’s Clinants

It is best not to think in terms of the dihedral angles but rather the dihedral clinants.
Namely e2Iθ is the clinant associated to the angle θ.

z
c
abc=1

a                     b
0                             1                                     Scissors Congruence: The Birth of Hyperbolic Volume – p.41/70
Ideal Tetrahedron’s Clinants

The compactiﬁcation of the space of ideal tetrahedra is all clinants triples (a, b, c) such
that abc = 1, "blown up" at (1, 1, 1). To see this, note that the z coordinate equals 1−a .
1−¯
b

z
c
abc=1

a                    b
0                            1                                   Scissors Congruence: The Birth of Hyperbolic Volume – p.42/70
Decomposing Ideal Tetrahedron

8
c
b
a
P

Scissors Congruence: The Birth of Hyperbolic Volume – p.43/70
Decomposing Ideal Tetrahedron

and double it.

c
b
a
P

Scissors Congruence: The Birth of Hyperbolic Volume – p.44/70
Decomposing Ideal Tetrahedron

Firepole this doubled ideal tetrahedron.

c
b
a

Scissors Congruence: The Birth of Hyperbolic Volume – p.45/70
Decomposing Ideal Tetrahedron

Then we have our 2-3 move which....

c
b
a

Scissors Congruence: The Birth of Hyperbolic Volume – p.46/70
Decomposing Ideal Tetrahedron

Then we have our 2-3 move which....

c
b
a

Scissors Congruence: The Birth of Hyperbolic Volume – p.47/70
Decomposing Ideal Tetrahedron

Then we have our 2-3 move which....

c
b
a

Scissors Congruence: The Birth of Hyperbolic Volume – p.48/70
Decomposing Ideal Tetrahedron

Then we have our 2-3 move which....

c
b
a

Scissors Congruence: The Birth of Hyperbolic Volume – p.49/70
Decomposing Ideal Tetrahedron

Then we have our 2-3 move which....

c
b
a

Scissors Congruence: The Birth of Hyperbolic Volume – p.50/70
Decomposing Ideal Tetrahedron

allows to view our double tetrahedron as three ideal tetrahedra.

c
b
a

Scissors Congruence: The Birth of Hyperbolic Volume – p.51/70
Decomposing Ideal Tetrahedron

From inﬁnity we see these three ideal tetrahedra are very special ideal tetrahedra, the
isosceles ideal tetrahedron.

−1/b

−1/a

2
b
a   2
c2
−1/a
−1/b
−1/c                −1/c

Scissors Congruence: The Birth of Hyperbolic Volume – p.52/70
The Isosceles Ideal Tetrahedron

Let us denote this isosceles ideal tetrahedron as II(a). We have just proved

IT (a, b, c) = II(a) + II(b) + II(c).

So we have reduced ﬁnding the volume of an ideal tetrahedron to ﬁnding the volume of
an isosceles ideal tetrahedron.

−1/a

−1/a                 2   =II(a)
a2                      a
−1/a
−1/a

Scissors Congruence: The Birth of Hyperbolic Volume – p.53/70
The Isosceles Ideal Tetrahedron

Equally important is that the z coordinate of an Isosceles ideal tetrahedron II(a) i s a
itself, and a z coordinate corresponds to an isosceles ideal tetrahedron if and only if it is
unit sized.

−1/a

−1/a                   2   =II(a)
a2                        a
−1/a
−1/a

Scissors Congruence: The Birth of Hyperbolic Volume – p.54/70
A Tetrahedron’s Root

Theorem:(Dupont,    Sah)
n
n                    ik2π
[z ] = n          [e     n    z]
k=1

Scissors Congruence: The Birth of Hyperbolic Volume – p.55/70
In particular

Corollary:(Kubert)
n
n                     ik2π
V ol(z ) = n         V ol(e     n       z)
k=1

Scissors Congruence: The Birth of Hyperbolic Volume – p.56/70
Milnor’s Theorem

Theorem:(Milnor)   A continuous function

f : S1 → R

that satisﬁes
z
f (z) = f (¯)

and
n
n
X                ik2π
f (z ) = n             f (e     n    z)
k=1

must be equal
c (Li2 (z)).

Li2 (ζ) is the Euler dilogarithm

Z       ζ   log(1 − s)
Li2 (ζ) =                          ds.
0           s

Scissors Congruence: The Birth of Hyperbolic Volume – p.57/70
The Birth of Volume

After normalizing, we have a formula due
to Lobachevski,

2V ol(IT (a, b, c)) =    (a) + (b) + (c).

Scissors Congruence: The Birth of Hyperbolic Volume – p.58/70
The Milnor Conjecture

Let
i2πp
M = spanQ {[e              q       ]}
and view the volume as a map, V ol, from
M to RQ. Conjecture: ker(V ol) is the Q
span of elements in the from
n
i2πp                    ik2π       i2πp
[e     q    ]−n         [e     n    e    nq     ]
k=1

Scissors Congruence: The Birth of Hyperbolic Volume – p.59/70
The Milnor conjecture

In words: all rational relations are conse-
quences of the Kubert identities.

Scissors Congruence: The Birth of Hyperbolic Volume – p.60/70
Dehn Invariant

Recall, Sis(H n ) ≡ <C> . Let us extend the Dehn invariant to <C> . If we have an ideal
T                                           T
points cut off with a horoball, we may use the cut off lengths to deﬁne
X
Dehn(P ) =         l(e) ⊗ θ(e).
e∈P

8
A

B
C

Scissors Congruence: The Birth of Hyperbolic Volume – p.61/70
Dehn Invariant

s Notice this is well deﬁned since if you use a different horosphere, then the difference
of our two candidate Dehn Invariants is
X
x⊗         θ = x ⊗ nπ = 0.
8                θ∈∞

x                    x
x
A
B        C

Scissors Congruence: The Birth of Hyperbolic Volume – p.62/70
Dehn Example?

There is no know explicit "Dehn counter example" in H 3 ! Below we have graphed
V ol(II(e2Iθ )), with respect to θ.

0                                  π
Scissors Congruence: The Birth of Hyperbolic Volume – p.63/70
Dehn Example?

We’d like (and expect) that every such ϕ(p/q) is irrational, and hence provides a "Dehn
counter example". But not one is known to be! We even have...

(p/q)π φ
Scissors Congruence: The Birth of Hyperbolic Volume – p.64/70
Dehn Example?

Theorem:(Dupont,Sah   ) If
Dehn(ϕ(1/N )) = 0

for any 1/N ∈ (0, 1/6), then the Milnor conjecture is false.

(p/q)π φ                                          Scissors Congruence: The Birth of Hyperbolic Volume – p.65/70
Dehn Kernel

Denote the kernel of Dehn restricted to
<C>
T    as D(C).
Notice: Dehn Sufﬁciency is equivalent to
(V ol, Dehn) being injective.
In other words that (V ol, Dehn) has trivial
kernel, or even more simply that V ol is
1-1 when restricted D(C).

Scissors Congruence: The Birth of Hyperbolic Volume – p.66/70
Countability Conjectures

Conjecture: V ol is 1-1 when restricted
D(C).
Conjecture: D(C) is countable.
Conjecture: dimQ (D(C)) > 1.

Scissors Congruence: The Birth of Hyperbolic Volume – p.67/70
Evidence

<C>
Theorem   (Suslin)    T    has the unique
division property.
Theorem: (Dupont, Sah) V ol(D(C)) is
countable.

Scissors Congruence: The Birth of Hyperbolic Volume – p.68/70
Suslin’s Theorem

The rectangle on top and the triangle below are both the middle polygon divided by 2.
Hence they are scissors congruent.

+

+
Scissors Congruence: The Birth of Hyperbolic Volume – p.69/70
Evidence

The unique division property say that for
every [P ] there exist a class
(1/n)[P ] ∈ Sis(H n ) and that if n[Q] = n[R]
then [Q] = [R]. Notice that
n
n                    ik2π
[z ] = n         [e     n    z]
k=1

is a candidate for division. Suslin showed
this candidate obeys the 2 − 3 relation.     Scissors Congruence: The Birth of Hyperbolic Volume – p.70/70

```
To top