Volumes of hyperbolic -manifolds. Notes on a paper of Gabai, Meyerhoff, and Milley

Authors:
T. H. Marshall and G. J. Martin

Journal:
Conform. Geom. Dyn. **7** (2003), 34-48

MSC (2000):
Primary 30F40, 30D50, 57M50

DOI:
https://doi.org/10.1090/S1088-4173-03-00081-X

Published electronically:
June 17, 2003

MathSciNet review:
1992036

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We present a new approach and improvements to the recent results of Gabai, Meyerhoff and Milley concerning tubes and short geodesics in hyperbolic -manifolds. We establish the following two facts: if a hyperbolic -manifold admits an embedded tubular neighbourhood of radius about any closed geodesic, then its volume exceeds that of the Weeks manifold. If the shortest geodesic of has length less than , then its volume also exceeds that of the Weeks manifold.

**1.**C. Adams,*The noncompact hyperbolic**-manifold of minimal volume*, Proc. Amer. Math. Soc.,**100**(1987), 601-606. MR**88m:57018****2.**I. Algol,*Volume change under drilling*, to appear.**3.**A. Beardon,*The geometry of discrete groups*, Springer-Verlag, 1981.**4.**C. Cao, F.W. Gehring and G.J. Martin,*Lattice constants and a lemma of Zagier*, Lipa's legacy (New York, 1995), 107-120, Contemp. Math.,**211**, Amer. Math. Soc., Providence, RI, 1997. MR**99a:30040****5.**C. Cao and R. Meyerhoff,*The orientable cusped hyperbolic**-manifolds of minimum volume*, Invent. Math.,**146**(2001), no. 3, 451-478. MR**2002i:57016****6.**D. Gabai, R. Meyerhoff and P. Milley,*Volumes of tubes in hyperbolic**-manifolds*, J. Differential Geom.,**57**(2001), no. 1, 23-46. MR**2002i:57017****7.**D. Gabai, R. Meyerhoff and N. Thurston,*Homotopy hyperbolic**-manifolds are hyperbolic*, Annals of Math., to appear.**8.**F. W. Gehring, T. H. Marshall and G.J. Martin,*On the spectrum of axial distances in Kleinian groups*, Indiana Math. J.,**47**(1998), 1-10. MR**2000b:30066****9.**F. W. Gehring and G. J. Martin,*Commutators, collars and the geometry of Möbius groups*, J. Anal. Math.,**63**(1994), 175-219. MR**96c:30040****10.**F. W. Gehring and G. J. Martin,*Precisely invariant collars and the volume of hyperbolic**-folds*. J. Differential Geom.,**49**(1998), no. 3, 411-435. MR**2000c:57030****11.**F. W. Gehring and G. J. Martin,*The volume of hyperbolic**-folds with**-torsion*, . Quart. J. Math. Oxford Ser.,**50**(1999), no. 197, 1-12. MR**2000c:57031****12.**F. W. Gehring, C. Maclachlan G. J. Martin and A. W. Reid,*Arithmeticity, Discreteness and Volume*, Trans. Amer. Math. Soc.,**349**(1997), 3611-3643. MR**98d:57022****13.**T. Jørgensen,*On discrete groups of Möbius transformations*, Amer. J. Math.,**78**(1976), 739-749. MR**55:658****14.**T. H. Marshall and G. J. Martin,*Cylinder packings in hyperbolic space*, preprint.**15.**R. Meyerhoff,*A lower bound for the volume of hyperbolic**-manifolds*Canadian J. Math.,**39**(1987), 1038-1056. MR**88k:57049****16.**A. Przeworski,*Tubes in hyperbolic**-manifolds*, Thesis. University of Chicago and Top. and Appl. 128/2-3, 103-122.**17.**A. Przeworski,*Density of tube packings in hyperbolic space*, to appear.

Retrieve articles in *Conformal Geometry and Dynamics of the American Mathematical Society*
with MSC (2000):
30F40,
30D50,
57M50

Retrieve articles in all journals with MSC (2000): 30F40, 30D50, 57M50

Additional Information

**T. H. Marshall**

Affiliation:
Department of Mathematics, University of Auckland, Auckland, New Zealand

Email:
t_marshall@math.auckland.ac.nz

**G. J. Martin**

Affiliation:
Department of Mathematics, University of Auckland, Auckland, New Zealand

Email:
martin@math.auckland.ac.nz

DOI:
https://doi.org/10.1090/S1088-4173-03-00081-X

Received by editor(s):
August 30, 2001

Received by editor(s) in revised form:
April 10, 2003

Published electronically:
June 17, 2003

Additional Notes:
Research supported in part by the N. Z. Marsden Fund and the N. Z. Royal Society (James Cook Fellowship)

Article copyright:
© Copyright 2003
American Mathematical Society