Undecidability and the developability of permutoids and rigid pseudogroups

TL;DR

This paper proves Cameron's conjecture that no algorithm can determine if a finite permutoid can be extended to a finite permutation group, and shows the existence problem for finite developments of rigid pseudogroups is unsolvable.

Contribution

It establishes the undecidability of key problems related to permutoids and rigid pseudogroups using recent results on the profinite triviality problem.

Findings

Cameron's conjecture is proven to be true.

The existence problem for finite developments of rigid pseudogroups is undecidable.

The work links permutoid developability to group-theoretic problems.

Abstract

A permutoid is a set of partial permutations that contains the identity and is such that partial compositions, when defined, have at most one extension in the set. In 2004 Peter Cameron conjectured that there can exist no algorithm that determines whether or not a finite permutoid based on a finite set can be completed to a finite permutation group, and he related this problem to the study of groups that have no non-trivial finite quotients. This note explains how our recent work on the profinite triviality problem for finitely presented groups can be used to prove Cameron's conjecture. We also prove that the existence problem for finite developments of rigid pseudogroups is unsolvable.