  Let $G$ be the group of complex points of a real semi-simple Lie group whose
fundamental rank is equal to 1, e.g. $G= \SL_2 (\C) \times \SL_2 (\C)$ or
$\SL_3 (\C)$. Then the fundamental rank of $G$ is $2,$ and according to the
conjecture made in \cite{BV}, lattices in $G$ should have 'little' --- in the
very weak sense of 'subexponential in the co-volume' --- torsion homology.
Using base change, we exhibit sequences of lattices where the torsion homology
grows exponentially with the \emph{square root} of the volume.
  This is deduced from a general theorem that compares twisted and untwisted
$L^2$-torsions in the general base-change situation. This also makes uses of a
precise equivariant 'Cheeger-M\"uller Theorem' proved by the second author
\cite{Lip1}.
