Rewriting the elements in the intersection of the kernels of two morphisms between free groups

TL;DR

This paper presents a method to rewrite elements in the intersection of kernels of two free group morphisms, assuming their kernel pairs permute, using concepts from higher covering theory of racks and quandles.

Contribution

It introduces a novel technique for rewriting elements in the intersection of kernels in free groups based on permutation conditions of kernel pairs.

Findings

Provides a method to express elements as products of generators in the intersection

Applies higher covering theory of racks and quandles to free group kernels

Assumes kernel pairs permute, such as in pushouts that are double extensions

Abstract

Let F be the free group functor, left adjoint to the forgetful functor between the category of groups GRP and the category of sets SET. Let f from A to B, and h from A to C be two functions in SET and let Ker(F(f)) and Ker(F(h)) be the kernels of the induced morphisms between free groups. Provided that the kernel pairs Eq(f) and Eq(h) of f and h permute (such as it is the case when the pushout of f and h is a double extension in SET), this short article describes a method to rewrite a general element in the intersection of Ker(F(f)) and Ker(F(g)) as a product of generators in A which is (f,h)-symmetric in the sense of the higher covering theory of racks and quandles.