Algebraicness of trace field of finite volume hyperbolic 3-manifold and dimension of $\mathrm{SL}(2,\mathbb{C})$-character variety Does the following statement:

"Let $G$ be a finitely generated
  group and let $X(G)$ be the
  $SL(2,\mathbb{C})$-character variety
  of $G$. Suppose $X(G)$ contains an
  irreducible component $X_0$ such that
  for every $g\in G$ and
  $\chi_{\rho}\in X_0$,  $\chi_{\rho}(g)$
  is an algebraic number. Then
  $\dim_{\mathbb{C}}X_0=0$."

contradict this next statement  which is  true by result of Thurston: 

"The trace field of a finite covolume
  Kleinian group is a finite extension
  of $\mathbb{Q}$"

I am asking because if we take the figure-8 complement then it is a complete finite volume hyperbolic 3-manifold  with finitely generated fundamental group. However the closure of the set of irreducible character has $\dim_{\mathbb{C}} =1 > 0$.
 A: As Misha said in the comments, if the dimension of the $\mathrm{SL}(2,\mathbb{C})$-character variety is greater than 0, there will be characters with values that are not algebraic numbers.  This is clear since traces of words in $G$ generate the coordinate ring of $X(G)$.  In particular, let $t_{g_1}$,...,$t_{g_m}$ be a generating set for the coordinate ring, where $t_{g_i}(\chi_\rho)=\mathrm{tr}(\rho(g_i))$.  Then the map $T:X(G)\to \mathbb{C}^m$ defined by $\chi_\rho\mapsto (...,t_{g_i}(\chi_\rho),...)$ is an algebraic embedding.  In particular, notice that for any $g\in G$ that $t_g(\chi_\rho)=\mathrm{tr}(\rho(g))=\chi_\rho(g)$, and so the image of the characters coincides with the image of the coordinates.
And the second statement appears in  Thurston's, The geometry and topology of 3-manifolds, Mimeographed lecture notes, Princeton University, 1978 as Proposition 6.7.4 according to Alan Reid's A note on trace-fields of Kleinian groups, Bull. London Math. Soc. 22 (1990), no. 4, 349–352, which is worth reading to see how to get an invariant of the commensurability class from the trace field.  As Misha states it is a consequence of Mostow Rigidity.
Again as Misha noted in the comments, there is no contradiction between these two statements. The Kleinian group $\Gamma$ is a discrete subgroup of
$\mathrm{PSL}(2,\mathbb{C})$ such that $H^3/\Gamma$ has finite volume, and the trace-field of $\Gamma$ is defined to be the field $\mathbb{Q}(\mathrm{tr}(\gamma)\ |\ \gamma \in \Gamma)$.  On the other hand, coordinates in the character variety are coordinates on the full moduli space of (unimodular) representations of $G$.  Not every point in the character variety corresponds to a discrete subgroup, let alone a hyperbolic structure.
