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Scissors Congruence The Birth of Hyperbolic Volume

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					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


We need one more decomposition. Start with an 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

				
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