Slicing up hyperbolic tetrahedra: from the infinite to the finite Yana Mohanty University of California, San Diego mohanty@math.ucsd.edu Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Overview Problem statement: Spherical 2-D geometry Hyperbolic 2-D geometry Hyperbolic 3-D geometry Hyperbolic tetrahedra Problem statement: Construct a finite tetrahadron out of ideal tetrahedra Overview Outline of method Motivation Scissors congruence problems Study of hyperbolic 3-manifolds Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Spherical geometry “Lines” are great circles Triangles are “plump” Any 2-dimensional map distorts angles and/or lengths “Lines” are great circles Each pair of lines intersects in two points Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

The Mercator projection: a conformal map of the sphere Angles shown are the true angles! (conformal) Areas near poles are greatly distorted Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

the “opposite” of spherical geometry Hyperbolic geometry: the “opposite” of spherical geometry Triangles are “skinny” Given a point P and a line L there are many lines through P that do not intersect L. Any 2-dimensional map distorts angles and/or lengths. A piece of a hyperbolic surface in space Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

The Poincare model of the hyperbolic plane Escher’s Circle Limit I lines Preserves angles (conformal) Distorts lengths Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Hyperbolic space LINES PLANES Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

The Poincare and upper half-space models (obtained by inversion) metric: z>0 metric: Inversion: z=0 Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

H3: The upper halfspace model (obtained by inversion) metric: “point at infinity” z>0 z=0 Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Lines and planes in the half-plane model of hyperbolic space Contains point at infinity lines PLANES Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Ideal tetrahedron in H3 (Poincare model) Convex hull of 4 points at the sphere at infinity Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Ideal tetrahedron in H3 (half-space model) B b B b g a a g C A C A Determined by triangle ABC View from above Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Hyperbolic tetrahedra ideal: 2 parameters ¾-ideal: 3 parameters finite: 6 parameters 1 or 2 ideal vertices also possible Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Problem statement: How do you make out of finitely many of these? The rules: an ideal tetrahedron may count as + or – use finitely many planar cuts Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

What is this needed for: part I Study of hyperbolic 3-manifolds 2-Manifold: An object which is homeomorphic to a plane near every one of its points. can be stretched into without tearing Example of a Euclidean 2-manifold A 2 manifold may NOT contain Can’t be stretched into a plane near this point Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Euclidean 3-manifold Example: 3-Torus Glue together opposite faces Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Hyperbolic 3-manifold Example: the Seifert-Weber space Drawing from Jeff Weeks’ Shape of Space Image by Matthias Weber Glue together opposite faces Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

A strange and amazing fact: The volume of a hyperbolic 3-manifold is a topological invariant (There is a continuous 1-1 map from X to Y with a continuous inverse) Homeomorphic 3-manifold X 3-manifold Y X and Y have the same volume Volume computation generally requires triangulating, that is, cutting up the manifold into tetrahedra. Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Triangulating a hyperbolic 2-manifold Finite hyperbolic octagon 2-holed torus glue Drawing by Tadao Ito In hyperbolic space triangulation involves finite tetrahedra (6-parameters) Better: express in terms of ideal tetrahedra (2-parameters) Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

What is this needed for: part II Solving scissors congruence problems in hyperbolic space: Given 2 polyhedra of equal volume, can one be cut up into a finite number of pieces that can be reassembled into the other one? Example in Euclidean space: “Hill’s tetrahedron” Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

An expression for volume that also gives a canonical decompositon? Exists for ideal tetrahedra: volume=L(a) volume=L(b) volume=L(g) (hidden) b a V =L(a)+L(b)+L(g), g finite! where is the Lobachevsky function. Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Construction of a 3/4-ideal tetrahedron out of ideal tetrahedra: extends “volume formula as a decomposition” idea to tetrahedra with finite vertex Algebraic Proved in 1982 by Dupont and Sah using homology. History: Geometric Mentioned as unknown by W. Neumann in 1998 survey article on 3-manifolds. Indications of construction given by Sah in 1981, but these were not well known. Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Main idea behind proof Make a a certain type of ¾-ideal tetrahedron first d d d p p rotated {a,b,c,p} = + - p c c c a a b b Inspiration for choosing ideal tetrahedra: another proof of Dupont and Sah Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Remainder of the proof Making a finite tetrahedron out of ¾-ideal tetrahedra Step 1: finite out of 1-ideal A B C D Step 2: 1-ideal out of ¾-ideal A B C E E C’ B’ E A B C D A’ ABCE=A’B’C’E-A’B’BE-B’C’CE-C’A’AE 1-ideal ideal ¾-ideal finite 1-ideal ABCD=ABCE-ABDE Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Summary Comparison of spherical and hyperbolic geometries Examples of conformal models Spherical: Mercator projection Hyperbolic: Poincare ball Introduced hyperbolic tetrahedra Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu

Summary, continued Constructing a finite tetrahedron out of ideal ones is helpful for studying -hyperbolic 3-manifolds volume is an invariant, so construction is helpful in the 3-dimensional equivalent of -scissors congruences want volume formula that is also a decomposition Main ingredient: constructing a certain ¾-ideal tetrahedron out of ideal tetrahedra. Idea comes from a proof by Dupont and Sah. Yana Mohanty, University of California, San Diego, mohanty@math.ucsd.edu