Inertial endomorphisms of an abelian group

TL;DR

This paper characterizes inertial endomorphisms of abelian groups, showing their structure, properties, and relations to automorphisms, especially under conditions like finite torsion-free rank.

Contribution

It provides a detailed description of inertial endomorphisms, their algebraic structure, and their relation to automorphisms in abelian groups with finite torsion-free rank.

Findings

Inertial endomorphisms form a ring containing multiplications and finitary endomorphisms.

Invertible inertial endomorphisms form a group when the group has finite torsion-free rank.

The group generated by inertial automorphisms is commutative modulo finitary automorphisms.

Abstract

We describe inertial endomorphisms of an abelian group $A$, that is endomorphisms $\varphi$ with the property $|(\varphi(X)+X)/X|<\infty$ for each $X\le A$. They form a ring containing multiplications, the so-called finitary endomorphisms and non-trivial instances. We show that inertial invertible endomorphisms form a group, provided $A$ has finite torsion-free rank. In any case, the group $IAut(A)$ they generate is commutative modulo the group $FAut(A)$ of finitary automorphisms, which is known to be locally finite. We deduce that $IAut(A)$ is locally-(center-by-finite). Also we consider the lattice dual property, that is that $|X/(X\cap \varphi(X))|<\infty$ for each $X\le A$. We show that this implies the above one, provided $A$ has finite torsion-free rank.