Explicit bounds for fixed subgroups of endomorphisms of free products

TL;DR

This paper investigates explicit bounds on the ranks of fixed subgroups of endomorphisms in free products, extending known results from free groups to more general algebraic structures.

Contribution

It provides new explicit bounds for the ranks of fixed subgroups of endomorphisms of free products, generalizing previous results from free groups.

Findings

Established explicit bounds for fixed subgroup ranks in free products

Extended Scott conjecture results from free groups to free products

Enhanced understanding of fixed subgroup structures in algebraic automorphisms

Abstract

For an automorphism $\phi$ of a free group $F_n$ of rank $n$, Bestvina and Handel showed that the rank $rk Fix(\phi)$ of the fixed subgroup is not greater than $n$ (the so-called Scott conjecture). Soon after Bestvina and Handel's announcement, their result was generalized by many authors in various directions. In this paper, we are interested in the fixed subgroups of endomorphisms of free products, focusing on explicit bounds for their ranks.