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Some amalgam structures for Bianchi groups

Authors: Charles Frohman and Benjamin Fine
Journal: Proc. Amer. Math. Soc. 102 (1988), 221-229
MSC: Primary 20E06,; Secondary 11F06,22E40,57M99
MathSciNet review: 920977
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Abstract: Splittings as an amalgamated sum, where one of the factors is the projective elementary matrices, are displayed for all the Bianchi groups except for the Euclidian ones. Also splittings as an HNN group, where the base is $ {\text{PS}}{{\text{L}}_2}(Z)$, are given for all the Bianchi groups except $ {\text{PS}}{{\text{L}}_2}({O_1})$ and $ {\text{PS}}{{\text{L}}_2}({O_3})$.

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  • [B1] L. Bianchi, Sui gruppi de sostituzioni lineari con coefficienti appartenenti a corpi quadratici imaginari, Math. Ann. 40 (1897), 332-412. MR 1510727
  • [B] M. Baker, Ramified primes and the homology of the Bianchi groups, Inst. Hautes Études Sci., preprint, 1982.
  • [Co] P. Cohn, On the structure of $ {\text{G}}{{\text{L}}_2}$ of a ring, Inst. Hautes Études Sci. Publ. Math. 30 (1966), 5-53. MR 0207856 (34:7670)
  • [CS] M. Culler and P. Shalen, Bounded separating surfaces in knot manifolds, Invent. Math. 75 (1984), 537-545. MR 735339 (85k:57010)
  • [Fi1] B. Fine, The structure of $ {\text{PS}}{{\text{L}}_2}(R)$, Ann. of Math. Studies, no. 79, Princeton Univ. Press, Princeton, N. J., 1974, pp. 145-170. MR 0352289 (50:4776)
  • [Fi2] -, The HNN and generalized free product structure of certain linear groups, Bull. Amer. Math. Soc. 81 (1975), 413-416. MR 0372059 (51:8276)
  • [Fl] D. Flöge, Zur Struktur der $ {\text{PS}}{{\text{L}}_2}$ über einegen imaginär-quadratischen Zahlringen, Math. Z. 183 (1983), 255-279. MR 704107 (85c:11043)
  • [GS] F. Grünewald and J. Schwermer, Arithmetic quotients of hyperbolic $ 3$-space, cusp forms and link complements, Duke Math J. 48 (1981), 183-200.
  • [H] A. Hatcher, Bianchi orbifolds of small discriminant, an informal report, Preprint.
  • [R] R. Riley, Applications of a computer implementation of Poincaré's theorem on fundamental polyhedra, Math. Comp. 40 (1983), 607-632. MR 689477 (85b:20064)
  • [Ro] J. Rohlfs, On the cuspidal cohomology of the Bianchi modular group, Math. Z. 188 (1985), 253-270. MR 772354 (86e:11042)
  • [S] D. Solitar, private correspondence.
  • [Sw] R. Swan, Generators and relations for certain special linear groups, Adv. in Math. 6 (1871), 1-78. MR 0284516 (44:1741)
  • [Th] W. Thurston, The geometry and topology of $ 3$-manifolds, mimeographed notes.
  • [V] K. Voightmann, Rational homology of Bianchi groups, Math. Ann. (to appear). MR 799670 (87a:22025)
  • [Z] R. Zimmert, Zur $ {\text{S}}{{\text{L}}_2}$ der ganzen Zahlen eines imaginär-quadratischen Zahlkörpers, Invent. Math. 19 (1973), 73-81. MR 0318336 (47:6883)
  • [Z1] H. Zieschang, Finite groups of mapping classes of surfaces, Lecture Notes in Math., vol. 875, Springer-Verlag, Berlin and New York, 1981. MR 643627 (86g:57001)

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Article copyright: © Copyright 1988 American Mathematical Society

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