  The cogrowth of a subgroup is defined as the growth of a set of coset
representatives which are of minimal length. A subgroup is essential if it
intersects non-trivially every non-trivial subgroup. The main result of this
paper is that every function $f:{\Bbb N}\cup \{0\}\rightarrow {\Bbb N}$ which
is strictly increasing, but at most exponential, is equivalent to a cogrowth
function of an essential subgroup of infinite index of the free group of rank
two. This class of functions properly contains the class of growth functions of
groups. The notions of growth and cogrowth of right ideals in algebras are
introduced. We show that when the algebra is without zero divisors then every
right ideal, whose cogrowth is less than that of the algebra, is essential.
