  An important problem in combinatorial noncommutative algebra is to
characterize the growth functions of finitely generated algebras (equivalently,
semigroups, or hereditary languages).
  The growth function of every finitely generated, infinite-dimensional algebra
is increasing and submultiplicative. The question of to what extent these
natural necessary conditions are also sufficient -- and in particular, whether
they are sufficient at least for sufficiently rapid functions -- was posed and
studied by various authors and has attracted a flurry of research.
  While every increasing and submultiplicative function is realizable as a
growth function up to a linear error term, we show that there exist arbitrarily
rapid increasing submultiplicative functions which are not equivalent to the
growth of any algebra, thus resolving the aforementioned problem and settling a
question posed by Zelmanov (and repeated by Alahmadi-Alsulami-Jain-Zelmanov).
These can be interpreted as `holes' in the space of growth functions,
accumulating to exponential functions in the order topology.
  We show that there exist monomial algebras and hereditary languages whose
growth functions encode the existence of non-prolongable words, and algebras
whose growth functions encode the existence of nilpotent ideals (in the graded
case). This negatively solves another conjecture of
Alahmadi-Alsulami-Jain-Zelmanov in the graded case.
