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Amalgamation and the invariant trace field of a Kleinian group. (English) Zbl 0728.57009

If \(\Gamma\) is a Kleinian group of finite covolume and \(\Gamma^{(2)}\) the subgroup generated by squares of elements of \(\Gamma\), then the invariant trace field k(\(\Gamma\)) of \(\Gamma\) is that generated over \({\mathbb{Q}}\) by the traces of elements in \(\Gamma^{(2)}\). This field is an invariant of the commensurability class of \(\Gamma\) [A. W. Reid, Bull. Lond. Math. Soc. 22, No.4, 349-352 (1990; Zbl 0706.20038)]. In the case where \(\Gamma\) is arithmetic, k(\(\Gamma\)) is the field of definition of the corresponding quaternion algebra A(\(\Gamma\)) and topological and geometric information on \(\Gamma\) can be related to algebraic and arithmetic information of k(\(\Gamma\)) and A(\(\Gamma\)). In this paper and in Proc. Res. Sem. Low-dim. Topology OSU 1990 (De Gruyter, 1991), the authors investigate relationships when \(\Gamma\) is no longer arithmetic. The main result here is that if \(\Gamma =\Gamma_{0^*H}\Gamma_ 1\) or \(\Gamma_{0^*H}\), where H is non-elementary, then \(k(\Gamma)=k(\Gamma_ 0).k(\Gamma_ 1)\) or \(k(\Gamma_ 0)\) respectively. As a consequence, the invariant tracefield is an invariant under mutation of the corresponding finite volume hyperbolic 3-manifold. The proof is obtained by relating these fields to certain fields obtained by considering the fixed points of the conjugacy class of a parabolic or loxodromic element. The main result holds without the restriction of finite covolume but does require that H be non-elementary.

MSC:

57M50 General geometric structures on low-dimensional manifolds
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations

Citations:

Zbl 0706.20038
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References:

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