Actions of Maximal Growth

TL;DR

This paper investigates acts and modules with maximal growth over free structures, revealing their near-free properties and compatibility with finiteness conditions, thus advancing understanding of their algebraic behavior.

Contribution

It introduces the concept of maximal growth in acts and modules over free structures and explores their properties and compatibility with finiteness conditions.

Findings

Maximal growth acts and modules are close to free structures.

Maximal growth can coexist with finiteness conditions.

Maximal growth implies specific algebraic properties.

Abstract

We study acts and modules of maximal growth over finitely generated free monoids and free associative algebras as well as free groups and free group algebras. The maximality of the growth implies some other specific properties of these acts and modules that makes them close to the free ones; at the same time, we show that being a strong "infiniteness" condition, the maximality of the growth can still be combined with various finiteness conditions, which would normally make finitely generated acts finite and finitely generated modules finite-dimensional.