Geodesic
Minimal co-volume hyperbolic lattices, I: The spherical points of a Kleinian group
Frederick W. Gehring1 ; Gaven J. Martin2
1 Department of Mathematics<br/>University of Michigan<br/>Ann Arbor, MI 48109<br/>United States
2 Institute of Information and Mathematical Sciences<br/>Massey University<br/>Albany Campus<br/>Private Bag 102-904<br/>North Shore Mail Centre<br/>Auckland<br/>New Zealand
Annals of mathematics, Tome 170 (2009) no. 1, pp. 123-161

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We identify the two minimal co-volume lattices of the isometry group of hyperbolic $3$-space that contain a finite spherical triangle group. These two groups are arithmetic and are in fact the two minimal co-volume lattices. Our results here represent the key step in establishing this fact, thereby solving a problem posed by Siegel in 1945. As a consequence we obtain sharp bounds on the order of the symmetry group of a hyperbolic $3$-manifold in terms of its volume, analogous to the Hurwitz $84g-84$ theorem of 1892.

DOI : 10.4007/annals.2009.170.123

Frederick W. Gehring 1 ; Gaven J. Martin 2

1 Department of Mathematics<br/>University of Michigan<br/>Ann Arbor, MI 48109<br/>United States
2 Institute of Information and Mathematical Sciences<br/>Massey University<br/>Albany Campus<br/>Private Bag 102-904<br/>North Shore Mail Centre<br/>Auckland<br/>New Zealand 
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Frederick W. Gehring; Gaven J. Martin. Minimal co-volume hyperbolic lattices, I:  The spherical points of a Kleinian group. Annals of mathematics, Tome 170 (2009) no. 1, pp. 123-161. doi: 10.4007/annals.2009.170.123

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