Growth rate for endomorphisms of finitely generated nilpotent groups and solvable groups

TL;DR

This paper investigates the growth rates of endomorphisms in finitely generated nilpotent and solvable groups, establishing a key equivalence for nilpotent groups and exploring algebraic properties in solvable groups.

Contribution

It generalizes the known result for automorphisms to endomorphisms in nilpotent groups and analyzes growth rates in specific solvable groups, including counterexamples.

Findings

Growth rate of endomorphism equals that on abelianization for nilpotent groups

Counterexample to a known result in solvable groups

Growth rate in solvable groups is an algebraic number

Abstract

We prove that the growth rate of an endomorphism of a finitely generated nilpotent group equals to the growth rate of induced endomorphism on its abelinization, generalizing the corresponding result for an automorphism in [14]. We also study growth rates of endomorphisms for specific solvable groups, lattices of Sol, providing a counterexample to a known result in [5] and proving that the growth rate is an algebraic number.