When Schrier transversals grow wild

Abstract

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.