y TtEnheints orientation note from Htt on 2T outward normals Def - PDF document

virtual Infinitely many geometric triangulations T Seminar 4121120 CUNY G Neil Hoffman Joint w Setting i hyperbolic 3 inflds 14 He 317 crisped manifolds noncompact volume finite orientable T Def a ideal geometric tetrahedron Tate is the of 4 hull convex coplanar points 2413 non on µ y TtEnheints orientation note from Htt on 2T outward normals Def a geometric ideal triangulation of M ideal tetrahedra is into decomp a geom reversing isometries along glued by mint their faces tear H

1980 Petronio 1990J Thurston Conjecture Every cusped hyperbolic 143 has a geometric ideal triangulation Whycaree Thurston's Dehn surgery theorem is way easier to prove if 14has geom triang Evidence Millions of examples leptin Penner 1988J Every cusped 143 a geometric ideal polyhedral decorup has idea pack HP by hookalls I premiages of caged a basin of attraction heroball each assign get tiling of 143 is polyhedral decomp dual En g µ Guehitand Schlemes 200832 or Dehn fillings of generic multi generic the EP decomp is a triangulation cusped 14

cusped decomp Scheimer Tillman Luo 2008 a finite Every cusped M has cover 19 Can be subdivided EP decomp whose geometric polyhedra into What can go wrong subdividing polyhedra in faces introduces diagonals They might be miosestent TATE 14 where every find cover a polyhedron P has all vertices at of A distinct C peripheralseparability cusps cusps of A order vertices order of every polyhedron to smallest vertex cone 53 Dodd Duan 20153 14 has infinitely many geometric related by 2 3 moves triangulations

tetra tetia 3 the bipyramid is as long convex as this be done geometrically operation can F theorem 2e 2e Hoffman Every cusped hyperbolic 3 manifold M has cover 19 admitting infinitely a finite many distinct geometric triangulations Ty where Tet is obtained Iz Ji by from 3 2 move a Proofouth backwardy cover 19 Step 2 M with 2 ind a geometric a distinguished cusp F J such triangulation T poking are only 2 tetrahedra that there T into AA ideal vertex each one

l T glued to T along 3 faces y 3 possible bipyramids I Isaiah more always possible cover of this picture also fine Reg J by subdoideing Epstein Penner decoup Get LST following peripheral separability Lind TY Steph M cover containing a a n n n oat resulting collection of cusps A Ao An decomp has only Epstein Penner a 2 1 cells into A ideal poking each vertex one A o Suffices to have unique I 1 shortest pathfrom A A UE A A A to Build this cover using doublecosetsep araality HE 7h2 are peripheral subgroup of If GE 7h2 f c ITM is an element want to separate IT M and G f H from 1 find F fintegp 4 IT M o t G f H ie n her 4 41 GfH n 4C 1 0 0

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