The isomorphism problem for plain groups is in $\Sigma_3^{\mathsf{P}}$

TL;DR

This paper proves that the isomorphism problem for plain groups, which are free products of finite groups and infinite cyclic groups, is in the complexity class a_3^P, advancing understanding of its computational complexity.

Contribution

The paper establishes that the isomorphism problem for plain groups can be decided within the a_3^P complexity class, using new algebraic and geometric characterizations.

Findings

The isomorphism problem for plain groups is in a_3^P.

New characterizations of groups presented by inverse-closed finite convergent rewriting systems.

Utilizes classical finite group isomorphism results of Babai and Szemer1edi.

Abstract

Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. S{\'e}nizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in $\mathsf{PSPACE}$ when the input is given in terms of so-called virtually free presentations. Here we consider the isomorphism problem for the class of \emph{plain groups}, that is, groups that are isomorphic to a free product of finitely many finite groups and finitely many copies of the infinite cyclic group. Every plain group is naturally and efficiently presented via an inverse-closed finite convergent length-reducing rewriting system. We prove that the isomorphism problem for plain groups given in this form lies in the polynomial time hierarchy, more precisely, in $\Sigma_3^{\mathsf{P}}$. This result is achieved by combining new geometric and algebraic characterisations of groups presented by inverse-closed finite convergent length-reducing rewriting systems developed in recent work of the second and third authors (2021) with classical finite group isomorphism results of Babai and Szemer\'edi (1984).