When Schreier transversals grow wild

The normalized cyclomatic quotient associated with presentations of finitely generated groups

“…, x n }, and let G be the associated Cayley graph. If H is the normal closure of R in F =< X > then it is shown in [8] that the growth function of G is equivalent to the rank-growth rk H of H. The rank-growth is defined by…”

“…Clearly definition (9) gives at least the same value as definition (8). To see that these definitions are equivalent we need to show that for every G ′ , G ′′ ∈ F (G) there exits H ∈ CF (G) such that G ′′ ⊆ H and ξ(G ′ ) ≤ ξ(H).…”