  We establish a number of results which say, roughly, that interpretation
functors preserve algebraic complexity.
  First we show that representation embeddings between categories of modules of
finite-dimensional algebras induce embeddings of lattices of pp formulas and
hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A
consequence is that the category of modules of any wild finite-dimensional
algebra has width $\infty$ and hence, if the algebra is countable, there is a
superdecomposable pure-injective representation.
  It is conjectured that a stronger result is true: that a representation
embedding from ${\rm Mod}\mbox{-}S$ to ${\rm Mod}\mbox{-}R$ admits an inverse
interpretation functor from its image and hence that, in this case, ${\rm
Mod}\mbox{-}R$ interprets ${\rm Mod}\mbox{-}S$. This would imply, for instance,
that every wild category of modules interprets the (undecidable) word problem
for (semi)groups. We show that the conjecture holds for finitely controlled
representation embeddings.
  Finally we prove that if $R,S$ are finite dimensional algebras over an
algebraically closed field and $I:{\rm Mod}\mbox{-}R\rightarrow{\rm
Mod}\mbox{-}S$ is an interpretation functor such that the smallest definable
subcategory containing the image of $I$ is the whole of ${\rm Mod}\mbox{-}S$
then, if $R$ is tame, so is $S$ and similarly, if $R$ is domestic, then $S$
also is domestic.
