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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