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Volumes of N ‐Cusped Hyperbolic 3‐Manifolds
Author(s) -
Adams Colin C.
Publication year - 1988
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-38.3.555
Subject(s) - mathematics , hyperbolic 3 manifold , finite volume method , cusp (singularity) , hyperbolic manifold , pure mathematics , manifold (fluid mechanics) , boundary (topology) , norm (philosophy) , tetrahedron , order (exchange) , ideal (ethics) , volume (thermodynamics) , hyperbolic triangle , mathematical analysis , hyperbolic group , geometry , hyperbolic function , physics , mechanical engineering , philosophy , finance , epistemology , quantum mechanics , political science , mechanics , law , engineering , economics
We prove that if M is an n ‐cusped hyperbolic 3‐manifold of finite volume, then vol ( M ) ⩾ nv , where v is the volume of an ideal regular tetrahedron in H 3 . This generalizes the case of n = 1 which was previously known. Stated another way, this says that the Gromov norm of an n ‐cusped hyperbolic 3‐manifold is at least n . We give all examples of manifolds for which this lower bound is realized when n = 1 or 2 and then prove that it cannot be realized for n ⩾ 3. Maximal cusp volumes in manifolds of low volume are then discussed followed by applications of these results to periods of homeomorphisms of finite order on 3‐manifolds with toral boundary components.