Shadowing for actions of some finitely generated groups

TL;DR

This paper introduces a shadowing property for finitely generated group actions, proves a shadowing lemma for nilpotent groups, and explores how hyperbolicity influences shadowing in linear actions, highlighting differences among various group types.

Contribution

It defines the shadowing property for group actions, proves a key lemma for nilpotent groups, and demonstrates the property’s dependence on hyperbolicity in linear actions of certain groups.

Findings

Shadowing property formulated for finitely generated groups.

Shadowing lemma established for nilpotent group actions.

Linear actions of non-abelian free groups lack shadowing.

Abstract

We introduce a notion of shadowing property for actions of finitely generated groups and study its basic properties. We formulate and prove a shadowing lemma for actions of nilpotent groups. We construct an example of a faithful linear action of a solvable Baumslag-Solitar group and show that the shadowing property depends on quantitative characteristics of hyperbolicity. Finally we show that any linear action of a non-abelian free group does not have the shadowing property.