Growth rate of an endomorphism of a group

TL;DR

This paper investigates the group-theoretic properties of the growth rate of endomorphisms in finitely generated groups, establishing bounds, relationships with subgroups and quotients, and computing growth rates for various group classes.

Contribution

It provides new bounds and formulations for the growth rate of endomorphisms, linking it to subgroup chains and extending calculations to specific group classes.

Findings

Growth rate is finite and bounded by generator image length.

Established equivalence with subgroup chain growth.

Computed growth rates for abelian and nilpotent groups.

Abstract

In [B] Bowen defined the growth rate of an endomorphism of a finitely generated group and related it to the entropy of a map $f:M \mapsto M$ on a compact manifold. In this note we study the purely group theoretic aspects of the growth rate of an endomorphism of a finitely generated group. We show that it is finite and bounded by the maximum length of the image of a generator. An equivalent formulation is given that ties the growth rate of an endomorphism to an increasing chain of subgroups. We then consider the relationship between growth rate of an endomorphism on a whole group and the growth rate restricted to a subgroup or considered on a quotient.We use these results to compute the growth rates on direct and semidirect products. We then calculate the growth rate of endomorphisms on several different classes of groups including abelian and nilpotent.