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Deformations of reducible representations of 3-manifold groups into \(\text{PSL}_2(C)\). (English) Zbl 1082.57007

Let \(M\) be a 3-manifold with torus boundary which is a rational homology sphere. Let \(\alpha:\pi_1(M)\rightarrow\mathbb C^*\) be a homomorphism and \(\rho_\alpha:\pi_1(M)\rightarrow \text{PSL}_2(\mathbb C)\) be the abelian representation given by \(\rho_\alpha(\gamma)=\pm \text{diag} (\alpha^{1\over2}(\gamma),\alpha^{-{1\over2}}(\gamma))\), \(\gamma\in\pi_1(M)\).
In this paper, the authors give a partial answer to the question: when can \(\rho_\alpha\) be deformed into irreducible representations? Their answer is a generalization of the results of M. Heusener, J. Porti and E. Suarez Peiro [J. Reine Angew. Math. 530, 191–227 (2001; Zbl 0964.57006)] where they considered only representations \(\alpha:\pi_1(M)\rightarrow\mathbb C^*\) which factor through \(H_1(M,\mathbb Z)/ \text{tors}(H_1(M,\mathbb Z))\) and for which \(\alpha^{1\over2}\) can be chosen as a homomorphism. Moreover, the authors have removed the condition that \(\rho_\alpha\) is not \(\partial\)-trivial from [loc. cit.].
It is to note that the approach given in this paper for the cohomological computations needed for the deformation and for the analysis of the tangent space is completely self contained and simplifies in several aspects the computations from [loc. cit.].

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
20C99 Representation theory of groups

Citations:

Zbl 0964.57006
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References:

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