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Zbl 1008.11015
Chinburg, Ted; Friedman, Eduardo; Jones, Kerry.N.; Reid, Alan W.
The arithmetic hyperbolic $3$-manifold of smallest volume. (English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 30, No.1, 1-40 (2001). ISSN 0391-173X

In search of the arithmetic hyperbolic $3$-manifold of smallest volume, the authors look at a conjecture stating that the Weeks manifold, which is obtained by $(5,1)$, $(5,2)$-Dehn surgery on the complement of the Whitehead link in $S^3$, is the arithmetic hyperbolic 3-manifold of smallest volume. In fact they prove that, with a volume of 0,9427073627769\dots, it is indeed the smallest one. They show that the arithmetic hyperbolic 3-manifold having the next smallest volume is the Meyerhoff manifold, obtained by $(5,1)$-surgery on the figure eigt knot complement. Its volume is 0,9813688288922\dots{} It is also shown that the closed hyperbolic 3-manifold of smallest volume has $b_1$ less than or equal to 2, where $b_1$ is the rank of the first cohomology with underlying field $Q$. Also nine orbifolds covered by arithmetic hyperbolic $3$-manifolds with volume less than or equal to 1 are listed.
[I.N.Cangül (Bursa)]

MSC 2000:: *11F06 Structure of modular groups and generalizations
57M99 Low-dimensional topology

Keywords: Weeks manifold; Meyerhoff manifold

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