  Schreier formula for the rank of a subgroup of finite index of a finitely
generated free group $F$ is generalized to an arbitrary (even infinitely
generated) subgroup $H$ through the Schreier transversals of $H$ in $F$. The
rank formula may also be expressed in terms of the cogrowth of $H$. We
introduce the rank-growth function $rk_H(i)$ of a subgroup $H$ of a finitely
generated free group $F$. $rk_H(i)$ is defined to be the rank of the subgroup
of $H$ generated by elements of length less than or equal to $i$ (with respect
to the generators of $F$), and it equals the rank of the fundamental group of
the subgraph of the cosets graph of $H$, which consists of the paths starting
at $1$ that are of length $\leq i$. When $H$ is supnormal, i.e. contains a
non-trivial normal subgroup of $F$, we show that its rank-growth is equivalent
to the cogrowth of $H$. A special case of this is the known result that a
supnormal subgroup of $F$ is of finite index if and only if it is finitely
generated. In particular, when $H$ is normal then the growth of the group
$G=F/H$ is equivalent to the rank-growth of $H$. A Schreier transversal forms a
spanning tree of the cosets graph of $H$, and thus its topological structure is
of a contractible spanning subcomplex of a simplicial complex. The
$d$-dimensional simplicial complexes that contain contractible spanning
subcomplexes have the homotopy type of a bouquet of $r$ $d$-spheres. When these
complexes are also $n$-regular then $r$ can be computed by generalizing the
rank formula (which applies to Schreier transversals) to higher dimensions.
