Schreier rewriting beyond the classical setting

TL;DR

This paper extends Schreier formulas to free monoids, associative algebras, and infinite index subgroups, providing new algebraic relations and applications in large modules and group theory.

Contribution

It generalizes Schreier formulas beyond classical groups to free monoids, associative algebras, and infinite index subgroups, introducing new algebraic relations.

Findings

Derived Schreier-type formulas for free monoids and associative algebras.

Established relations for ranks of subactions and intersections.

Applied large modules over free associative algebras in group theory contexts.

Abstract

Using actions of 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 the paper Olshanskii, A. Yu.; Osin, D.V., Large groups and their periodic quotients, Proc. Amer. Math. Soc., 136 (2008), 753 - 759.