Schreier rewriting beyond the classical setting

Abstract

Using actions of the free monoids and free associative algebras, we establish some Schreier-type formulas involving the ranks of actions and the ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish the generalization of the Schreier formula to the case of subgroups of infinite index. We also study and apply large modules over free associative algebras in the spirit of [1,11].

Actions of maximal growth

“…If none of (a) or (b) applies, then Γ i = Γ i−1 = Γ. It is easy to check that the chain thus constructed satisfies all conditions (1) to (3) and that Γ is the union of all Γ i .…”

“…According to [4], any such module has a free submodule of finite codimension, in particular, such module is large. Given an algebra R over a field Φ, a right R-module M is called large if it has a submodule of finite codimension which can be mapped onto the free R-module R. Free actions in all four cases and large module in the case of A r and F r have been studied in our previous paper [3].…”