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Fuchsian Subgroups of Bianchi Groups
Author(s):
D.
G.
James;
C.
Maclachlan
Journal:
Trans. Amer. Math. Soc.
348
(1996),
1989-2002.
MSC (1991):
Primary 11F06;
Secondary 11E08, 11E12, 20H10.
MathSciNet review:
1348863
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Abstract:
A maximal non-elementary Fuchsian subgroup of a Bianchi group  has an invariant circle or straight line under its linear fractional action on the complex plane, to which is associated a positive integer  , the discriminant, which, in turn, is an invariant of the wide commensurability class of the Fuchsian subgroup. In this paper, for all Bianchi groups, we classify the conjugacy classes of these maximal Fuchsian subgroups by determining the number with given discriminant.
References:
- 1.
- A. Borel, Commensurability classes and volumes of hyperbolic 3-manifolds, Ann. Sci. Norm. Pisa 8 (1981), 1 - 33. MR 82j:22008
- 2.
- J. Elstrodt, F. Grunewald and J. Mennicke, Discontinuous groups on three-dimensional hyperbolic space : analytical theory and arithmetic applications, Russian Math. Surveys 38 (1983), 137 - 168. MR 87c:11053
- 3.
- A.G. Earnest and J.S. Hsia, Spinor norms of local integral rotations II, Pacific J. Math. 61 (1975), 71 - 86. MR 53:7946
- 4.
- B. Fine, The algebraic theory of the Bianchi groups, Marcel - Dekker (1989). MR 90h:20002
- 5.
- D.G. James and S.M. Rosenzweig, Associated vectors in lattices over valuation rings, Amer. J. Math. 90 (1968), 295 - 307. MR 36:3754
- 6.
- D.G. James Integral sums of squares in algebraic number fields, Amer. J. Math. 113 (1991), 129 - 146. MR 92j:11036
- 7.
- C. Maclachlan, Fuchsian subgroups of the groups
 , Low-Dimensional Topology and Kleinian groups, ed. D.B.A. Epstein, L.M.S. Lecture Notes Series 112 (1987), 251 - 258. MR 89a:11049 - 8.
- C. Maclachlan and A.W. Reid, Parametrizing Fuchsian Subgroups of the Bianchi Groups, Canadian J. Math. 43 (1991), 158 - 181. MR 92d:11040
- 9.
- C. Maclachlan and A.W. Reid, The arithmetic structure of tetrahedral groups of hyperbolic isometries, Mathematika 36 (1989), 221 - 240. MR 91b:11055
- 10.
- O.T. O'Meara, Introduction to Quadratic Forms, Springer - Verlag (1963). MR 27:2485
- 11.
- T. Tamagawa, On the structure of orthogonal groups, Amer. J. Math. 80 (1958), 191 - 197. MR 20:71
- 12.
- L. Ya. Vulakh, Classification of Fuchsian subgroups of some Bianchi groups, Canadian Math. Bull. 34 (1991), 417 - 422. MR 92i:11047
- 13.
- L. Ya. Vulakh, Maximal Fuchsian subgroups of extended Bianchi groups, Number Theory with an Emphasis on the Markoff Spectrum, ed. A. Pollington Marcel - Dekker (1993), 297 - 310. MR 94g:11028
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Additional Information:
D.
G.
James
Affiliation:
Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email:
james@math.psu.edu
C.
Maclachlan
Affiliation:
Department of Mathematical Sciences, University of Aberdeen, Old Aberdeen, Aberdeen AB9 2TY, Scotland
Email:
cmac@maths.aberdeen.ac.uk
DOI:
10.1090/S0002-9947-96-01606-6
PII:
S 0002-9947(96)01606-6
Received by editor(s):
November 11, 1994
Received by editor(s) in revised form:
July 6, 1995
Additional Notes:
The first author partially supported by NSA and NSF grants
Both authors would like to thank the Mathematics Department of the University of Auckland for its hospitality
Copyright of article:
Copyright
1996,
American Mathematical Society
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