  We consider the asymptotics of the Turaev-Viro and the Reshetikhin-Turaev
invariants of a hyperbolic $3$-manifold, evaluated at the root of unity
$\exp({2\pi\sqrt{-1}}/{r})$ instead of the standard $\exp({\pi\sqrt{-1}}/{r})$.
We present evidence that, as $r$ tends to $\infty$, these invariants grow
exponentially with growth rates respectively given by the hyperbolic and the
complex volume of the manifold. This reveals an asymptotic behavior that is
different from that of Witten's Asymptotic Expansion Conjecture, which predicts
polynomial growth of these invariants when evaluated at the standard root of
unity. This new phenomenon suggests that the Reshetikhin-Turaev invariants may
have a geometric interpretation other than the original one via $SU(2)$
Chern-Simons gauge theory.
