Vertex links and the Grushko decomposition

TL;DR

This paper presents a polynomial-time algorithm for computing the Grushko decomposition of fundamental groups of specific graphs of free groups, utilizing vertex link analysis of associated CAT(0) square complexes.

Contribution

It introduces a novel polynomial-time method leveraging vertex link analysis to decompose groups, advancing computational techniques in geometric group theory.

Findings

Algorithm operates in polynomial time

Transforms complex CAT(0) square complexes to simpler forms

Ensures vertex links meet strong connectivity conditions

Abstract

We develop an algorithm of polynomial time complexity to construct the Grushko decomposition of fundamental groups of graphs of free groups with cyclic edge groups. Our methods rely on analysing vertex links of certain CAT(0) square complexes naturally associated with a special class of the above groups. Our main result transforms a one-ended CAT(0) square complex of the above type to one whose vertex links satisfy a strong connectivity condition, as first studied by Brady and Meier.