VIEWS: 10 PAGES: 70 POSTED ON: 12/23/2009
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 q 1 z=[p,q;r,s] r 8 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 1/(1−z) (z−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 w − zw z − zw − , [z] + [w] − [zw] − 1 − zw 1 − zw where z and w are complex numbers, together 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 ) = T n 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 removing 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 understanding the volume of an ideal tetrahedron. 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. A B C 8 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. A B 8 C 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 0 A B 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 0 a b 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 0 a b 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 a P b Scissors Congruence: The Birth of Hyperbolic Volume – p.43/70 Decomposing Ideal Tetrahedron and double it. c a P b Scissors Congruence: The Birth of Hyperbolic Volume – p.44/70 Decomposing Ideal Tetrahedron Firepole this doubled ideal tetrahedron. c a b Scissors Congruence: The Birth of Hyperbolic Volume – p.45/70 Decomposing Ideal Tetrahedron Then we have our 2-3 move which.... c a b Scissors Congruence: The Birth of Hyperbolic Volume – p.46/70 Decomposing Ideal Tetrahedron Then we have our 2-3 move which.... c a b Scissors Congruence: The Birth of Hyperbolic Volume – p.47/70 Decomposing Ideal Tetrahedron Then we have our 2-3 move which.... c a b Scissors Congruence: The Birth of Hyperbolic Volume – p.48/70 Decomposing Ideal Tetrahedron Then we have our 2-3 move which.... c a b Scissors Congruence: The Birth of Hyperbolic Volume – p.49/70 Decomposing Ideal Tetrahedron Then we have our 2-3 move which.... c a b Scissors Congruence: The Birth of Hyperbolic Volume – p.50/70 Decomposing Ideal Tetrahedron allows to view our double tetrahedron as three ideal tetrahedra. c a b 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 b −1/b −1/c 2 2 a c2 −1/a −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 a2 −1/a −1/a a 2 =II(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 a2 −1/a −1/a a 2 =II(a) Scissors Congruence: The Birth of Hyperbolic Volume – p.54/70 A Tetrahedron’s Root Theorem:(Dupont, n Sah) n [z ] = n k=1 [e ik2π n z] Scissors Congruence: The Birth of Hyperbolic Volume – p.55/70 In particular Corollary:(Kubert) n V ol(z ) = n k=1 n V ol(e ik2π n z) Scissors Congruence: The Birth of Hyperbolic Volume – p.56/70 Milnor’s Theorem Theorem:(Milnor) A continuous function f : S1 → R that satisﬁes f (z) = f (¯) z and f (z ) = n must be equal c (Li2 (z)). Li2 (ζ) is the Euler dilogarithm Li2 (ζ) = Z ζ 0 n n X f (e ik2π n z) k=1 log(1 − s) ds. 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 M = spanQ {[e i2πp 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 [e i2πp q n ]−n k=1 [e ik2π n e i2πp nq ] Scissors Congruence: The Birth of Hyperbolic Volume – p.59/70 The Milnor conjecture In words: all rational relations are consequences 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 Dehn(P ) = X l(e) ⊗ θ(e). e∈P A B C 8 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. θ∈∞ x x 8 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> T Theorem (Suslin) 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 [z ] = n k=1 n [e ik2π n z] is a candidate for division. Suslin showed this candidate obeys the 2 − 3 relation. Scissors Congruence: The Birth of Hyperbolic Volume – p.70/70