-torsion and hyperbolic volume

Authors:
F. W. Gehring and G. J. Martin

Journal:
Proc. Amer. Math. Soc. **117** (1993), 727-735

MSC:
Primary 30F40; Secondary 20H10, 57S30

MathSciNet review:
1116260

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The Kleinian group is shown to have minimal covolume among all Kleinian groups containing torsion of order (the associated hyperbolic orbifold is also the minimal volume cusped orbifold). This follows from: Any cocompact Kleinian group with torsion of order has covolume at least . As a consequence, any compact hyperbolic manifold with a symmetry of order (with fixed points) has volume at least . These results follow from new collaring theorems for torsion in a Kleinian group arising from our generalizations of the Shimizu-Leutbecher inequality.

**[1]**Colin C. Adams,*The noncompact hyperbolic 3-manifold of minimal volume*, Proc. Amer. Math. Soc.**100**(1987), no. 4, 601–606. MR**894423**, 10.1090/S0002-9939-1987-0894423-8**[2]**Alan F. Beardon,*The geometry of discrete groups*, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR**698777****[3]**F. W. Gehring and G. J. Martin,*Stability and extremality in Jørgensen’s inequality*, Complex Variables Theory Appl.**12**(1989), no. 1-4, 277–282. MR**1040927****[4]**F. W. Gehring and G. J. Martin,*Inequalities for Möbius transformations and discrete groups*, J. Reine Angew. Math.**418**(1991), 31–76. MR**1111201****[5]**-,*Discreteness in Kleinian groups and iteration theory*(to appear).**[6]**-,*Commutators, collars and the geometry of Möbius groups*(to appear).**[7]**-,*Volume and torsion in hyperbolic*-*folds*(to appear).**[8]**Troels Jørgensen,*On discrete groups of Möbius transformations*, Amer. J. Math.**98**(1976), no. 3, 739–749. MR**0427627****[9]**S. Kojima and Y. Miyamoto,*The smallest hyperbolic*-*manifold with totally geodesic boundary*(to appear).**[10]**Bernard Maskit,*Kleinian groups*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR**959135****[11]**Bernard Maskit,*Some special 2-generator Kleinian groups*, Proc. Amer. Math. Soc.**106**(1989), no. 1, 175–186. MR**937850**, 10.1090/S0002-9939-1989-0937850-4**[12]**Robert Meyerhoff,*A lower bound for the volume of hyperbolic 3-manifolds*, Canad. J. Math.**39**(1987), no. 5, 1038–1056. MR**918586**, 10.4153/CJM-1987-053-6**[13]**Robert Meyerhoff,*The cusped hyperbolic 3-orbifold of minimum volume*, Bull. Amer. Math. Soc. (N.S.)**13**(1985), no. 2, 154–156. MR**799800**, 10.1090/S0273-0979-1985-15401-1**[14]**Robert Meyerhoff,*Sphere-packing and volume in hyperbolic 3-space*, Comment. Math. Helv.**61**(1986), no. 2, 271–278. MR**856090**, 10.1007/BF02621915**[15]**W. P. Thurston,*The geometry and topology of*-*manifolds*, Princeton Univ. Lecture notes, 1980.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
30F40,
20H10,
57S30

Retrieve articles in all journals with MSC: 30F40, 20H10, 57S30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1993-1116260-4

Article copyright:
© Copyright 1993
American Mathematical Society