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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 109, Issue 3
  • May 1991, pp. 509-515

Amalgamation and the invariant trace field of a Kleinian group

  • Walter D. Neumann (a1) and Alan W. Reid (a1)
  • DOI:
  • Published online: 24 October 2008

Let Γ be a Kleinian group of finite covolume and denote by Γ(2) the subgroup generated by {γ2:γ ∈ Γ}. In [9] the trace field of Γ(2) was shown to be an invariant of the commensurability class of Γ. In [8] this field was termed the invariant trace field of Γ and further properties of this field were studied. Following the notation of [8] we denote the invariant trace field of Γ by k(Γ).

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[1]C. Adams . Thrice punctured spheres in hyperbolic 3-manifolds. Trans. Amer. Math. Soc. 287 (1985), 645656.

[2]G. Faltings . Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73 (1983), 349366;

Invent. Math. 75 (1984), 381.

[3]S. P. Kerckhoff . The Neilsen Realization Problem. Ann. of Math. (2) 117 (1983), 235265.

[4]W. B. R. Lickorish and K. C. Millet . A polynomial invariant of oriented links. Topology 26 (1987), 107141.

[5]A. M. Macbeath . Commensurability of cocompact three-dimensional hyperbolic groups. Duke Math. J. 50 (1983), 12451253.

[9]A. W. Reid . A note on trace-fields of Kleinian groups. Bull. London Math. Soc. 22 (1990), 349352.

[10]D. Ruberman . Mutations and volumes of links in S3. Invent. Math. 90 (1987), 189215.

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