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Is there an example of an irreducible and boundary irreducible $3$-manifold $M$ with torus boundary and a non-abelian representation $\rho: \pi_1(M) \to \mathrm{SL}_2(\mathbb{C})$, a non-constant analytic arc $\chi_t$ of irreducible representations with endpoint at $\chi_{\rho}$ in the $\mathrm{SL}_2(\mathbb{C})$ character variety $X(M)$ of $\pi_1(M)$, satisfying the condition:

  • for every infinite sequence $\{u_1,u_2,\cdots,u_n,\cdots\}$ of elements $u_i\in Z^1(\pi_1(M),Ad_{\rho})$, the expression $$\rho_t=\exp(u_1 t+ u_2 t^2+ \cdots + u_n t^n +\cdots) \rho$$ does not define a lift of $\chi_t$ in $\overline {R(M)^{irr}} \subset \mathrm{Hom}(\pi_1(M),\mathrm{SL}_2(\mathbb{C}))$.
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    $\begingroup$ The first sentence of your question is difficult to parse -- you might consider splitting it into shorter sentences. $\endgroup$
    – Stefan Kohl ♦
    Commented Oct 16, 2015 at 13:05
  • $\begingroup$ You might have a look at Fine, Kirk, and Klassen, A local analytic splitting of the holonomy map on flat connections. Math. Ann. 299 (1994), no. 1, 171–189. $\endgroup$
    – Danny Ruberman
    Commented Jun 26, 2016 at 11:43

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