When Schreier Transversals Grow Wild

We introduce the rank-growth function rkH(i) of a subgroup H of a finitely generated free group F . rkH(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 ≤ 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.