Computer-assisted methods in Sigma-theory

TL;DR

This paper introduces an algorithm for recognizing character membership in Sigma-theory groups, relying on group properties and solvability of the word problem, with implications for computational group theory.

Contribution

It presents a novel algorithm that determines character membership in Sigma-theory groups using finite data and specific group properties.

Findings

Algorithm effectively recognizes Sigma-m group characters.

Finite data suffices for membership proof in a neighborhood.

Applicability depends on group being of type FP_m or F_2 and having a solvable word problem.

Abstract

We develop an algorithm for recognizing whether a character belongs to $\Sigma^m$. In order to apply it we just need to know that the ambient group is of type $\mathrm{FP}_m$ or of type $\mathrm{F}_2$ and that the word problem is solvable for this group. Then finite data is sufficient proof of membership in $\Sigma^m$, not just for the given character but also for a neighborhood of it.