On Eigenvalues of Free Group Endomorphisms

TL;DR

This paper investigates the eigenvalues associated with endomorphisms of free groups, showing that the nonzero eigenvalues are preserved when restricted to finite-index invariant subgroups, revealing structural properties of these mappings.

Contribution

It establishes that the nonzero eigenvalues of a free group endomorphism are contained within those of its restriction to finite-index invariant subgroups, a novel insight into their spectral behavior.

Findings

Nonzero eigenvalues are preserved under restriction to finite-index invariant subgroups.

Eigenvalues of the endomorphism are contained in the eigenvalues of its restriction.

Provides a new understanding of the spectral properties of free group endomorphisms.

Abstract

Let \phi be an endomorphism of a finitely generated free group F, and let H be a finite-index subgroup of F that is invariant under \phi. The nonzero eigenvalues of \phi are contained in the eigenvalues of \phi restricted to H.