Let $X$
be a complete finite-volume orientable hyperbolic $3$-manifold,
and let $\Gamma$
be a Kleinian representation of $\pi\_1(X)$.
Let $K\Gamma:=\mathbb{Q}\big(\{\mathrm{tr}\mid\gamma\in\Gamma\}\big)$,
i.e. its *trace field*.
Let $\Gamma^{(2)}:=\langle\gamma^2\mid\gamma\in\Gamma\rangle$,
the group generated by squares in $\Gamma$
and let $k\Gamma=K\Gamma^{(2)}$,
i.e. its *invariant trace field*.
Then these are number fields,
$K\Gamma$
is a manifold invariant and $k\Gamma$
is a commensurability invariant.


I've been told that if $X$
contains a immersed closed totally-geodesic surface,
then $K\Gamma$
is a degree $2$
extension of a real field $F$
(where $F$
is the trace field of the surface
under a fixed embedding into $X$).


Is this correct?
Do they really mean to say this of $k\Gamma$
rather than of $K\Gamma$?
Are there some additional conditions I'm missing that do make this true for $K\Gamma$?


If this is true,
would you please provide a reference?
(I'm scouring through my old copy of Maclachlan and Reid and can't seem to find it.)
If it isn't true,
would you know a counterexample?




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**Edit:**
I think this is true if I add the requirement that $\Gamma$
be arithmetic. The only part of that I'm missing is that the trace field of the surface must be the full real part of the trace field of the $3$-manifold. I can't find a reference for that either though.


It's obvious (trivial even) in the arithmetic noncompact case.



Trace field of a hyperbolic $3$-manifold with a totally geodesic subsurface
The answer to this follows easily from section 9.5 of Machlachlan and Reid -- which somehow I managed not to see every other time I checked the book! The thing stated in my question is not true, but the following is true.


Suppose $X$
is arithmetic and contains an immersed closed totally-geodesic surface $S$.
Then $\Gamma$
contains a non-elementary Fuchsian group $\Delta'$.
Then $\Delta'$
is contained in an arithmetic Fuchsian group $\Delta$
which satisfies $[k\Gamma:k\Delta]=2$
and $k\Gamma\cap\mathbb{R}=k\Delta$.


