Electronic Research Announcements

Author(s): Brent Everitt; John Ratcliffe; Steven Tschantz
Journal: Electron. Res. Announc. Amer. Math. Soc. 11 (2005), 40-46.
MSC (1991): Primary 57M50
Posted: May 27, 2005
Retrieve article in: PDF DVI PostScript

Abstract: By gluing together copies of an all right-angled Coxeter polytope a number of open hyperbolic

-manifolds with Euler characteristic

are constructed. They are the first known examples of hyperbolic

-manifolds having the smallest possible volume.

1.: C. Adams, The noncompact hyperbolic -manifold of minimum volume, Proc. AMS 100 (1987), 601-606. MR 0894423 (88m:57018)
2.: M. Belolipetsky, On volumes of arithmetic quotients of , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear).
3.: R. Borcherds, Automorphism groups of Lorentzian lattices, J. Algebra 111 (1987), no. 1, 133-153. MR 0913200 (89b:20018)
4.: A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. 75 (1962), no. 3, 485-535. MR 0147566 (26:5081)
5.: T. Chinburg, E. Friedman, K. N. Jones, and A. W. Reid, The arithmetic hyperbolic 3-manifold of smallest volume, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 30 (2001), no. 4, 1-40. MR 1882023 (2003a:57027)
6.: I. M. Chiswell, The Euler characteristic of graph products and Coxeter groups, in Discrete Groups and Geometry, W. J. Harvey and Colin Maclachlan (Editors), London Math. Soc. Lect. Notes, vol. 173, 1992, pp. 36-46. MR 1196914 (94a:05090)
7.: I. M. Chiswell, Euler characteristics of groups, Math. Z. 147 (1976), 1-11. MR 0396785 (53:645)
8.: M. D. E. Conder and C. Maclachlan, Small volume compact hyperbolic -manifolds, to appear, Proc. Amer. Math. Soc.
9.: J. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Second Edition, Springer, 1993. MR 1194619 (93h:11069)
10.: J. Conway and N. J. A. Sloane, Leech roots and Vinberg groups, Proc. Roy. Soc. London Ser. A 384 (1982), no. 1787, 233-258. MR 0684311 (84b:10048)
11.: H. S. M. Coxeter, Discrete groups generated by reflections, Ann. of Math. 35 (1934), no. 2, 588-621. MR 1503182
12.: M. W. Davis, A hyperbolic 4-manifold, Proc. Amer. Math. Soc. 93 (1985), 325-328. MR 0770546 (86h:57016)
13.: B. Everitt, Coxeter groups and hyperbolic manifolds, Math. Ann. 330 (2004), no. 1, 127-150. MR 2091682
14.: B. Everitt, -Manifolds from Platonic solids, Top. App. 138 (2004), 253-263. 2035484 (2004m:57031)
15.: B. Everitt, J. Ratcliffe and S. Tschantz, Arithmetic hyperbolic -manifolds of smallest volume, (in preparation).
16.: V. Matveev and A. Fomenko, Constant energy surfaces of Hamilton systems, enumeration of three-dimensional manifolds in increasing order of complexity, and computation of volumes of closed hyperbolic manifolds, Russian Math. Surveys 43 (1988), 3-24. MR 0937017 (90a:58052)
17.: C. Cao and G. R. Meyerhoff, The orientable cusped hyperbolic -manifolds of minimum volume, Invent. Math. 146 (2001), no. 3, 451-478. MR 1869847 (2002i:57016)
18.: J. Milnor and D. Husemoller, Symmetric Bilinear Forms, Springer, 1973. MR 0506372 (58:22129)
19.: A. Neumaier and J. J. Seidel, Discrete Hyperbolic Geometry, Combinatorica 3 (1983), 219-237. MR 0726460 (85h:51025)
20.: I. Prok, Classification of dodecahedral space forms, Beiträge Algebra Geom. 39 (1998), no. 2, 497-515. MR 1642716 (99i:52029)
21.: J. Ratcliffe, Foundations of hyperbolic manifolds, Graduate Texts in Mathematics 149, Springer 1994. MR 1299730 (95j:57011)
22.: J. Ratcliffe and S. Tschantz, Volumes of integral congruence hyperbolic manifolds, J. Reine Angew. Math. 488 (1997), 55-78. MR 1465367 (99b:11076)
23.: J. Ratcliffe and S. Tschantz, The volume spectrum of hyperbolic 4-manifolds, Experiment. Math. 9 (2000), no. 1, 101-125. MR 1758804 (2001b:57048)
24.: J. Ratcliffe and S. Tschantz, Integral congruence two hyperbolic -manifolds, Geom. Dedicata 107 (2004), 187-209. MR 2110762
25.: J.-P. Serre, A Course in Arithmetic, Graduate Texts in Mathematics 7, Springer, 1973. MR 0344216 (49:8956)
26.: C. L. Siegel, Über die analytische Theorie der quadratischen Formen II, Ann. Math. 37 (1936), 230-263. MR 1503276
27.: E. B. Vinberg, Hyperbolic reflection groups, Uspekhi Mat. Nauk 40 (1985), no. 1, 29-66; English transl., Russian Math. Surveys 40 (1985), no. 1, 31-75. MR 0783604 (86m:53059)
28.: E. B. Vinberg and I. M. Kaplinskaya, On the groups ${O}_{18,1}(\mathbb Z)$ and ${O}_{19,1} (\mathbb Z)$ , Dokl. Akad. Nauk. SSSR 238 (1978), no. 6, 1273-1275; English transl., Soviet Math. Dokl. 19 (1978), no. 1, 194-197. MR 0476640 (57:16199)
29.: E. B. Vinberg, On the groups of units of certain quadratic forms, Math. Sb. 87 (1972), no. 1, 18-36; English transl., Math. USSR, Sb. 16 (1972), no. 1, 17-35. MR 0295193 (45:4261)
30.: J. Weeks, Hyperbolic structures on -manifolds, Ph.D. thesis, Princeton University, 1985.

Brent Everitt
Affiliation: Department of Mathematics, University of York, York YO10 5DD, England
Email: bje1@york.ac.uk

John Ratcliffe
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, TN 37240
Email: ratclifj@math.vanderbilt.edu

Steven Tschantz
Affiliation: Department of Mathematics, Vanderbilt University, Nashville, TN 37240
Email: tschantz@math.vanderbilt.edu

DOI: 10.1090/S1079-6762-05-00145-9
PII: S 1079-6762(05)00145-9
Received by editor(s): October 31, 2004
Posted: May 27, 2005
Additional Notes: The first author is grateful to the Mathematics Department, Vanderbilt University for its hospitality during a stay when the results of this paper were obtained.
Communicated by: Walter Neumann
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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