Dietrich, HeikoElder, MurrayPiggott, AdamQiao, YoumingWeiß, Armin2025-06-242025-06-2497839597722281868-8969ORCID:/0000-0002-9156-9096/work/162370434http://www.scopus.com/inward/record.url?scp=85127101310&partnerID=8YFLogxKhttps://hdl.handle.net/1885/733764658Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. Sénizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in PSPACE when the input is given in terms of so-called virtually free presentations. Here we consider the isomorphism problem for the class of 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 ΣP3. 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 Szemerédi (1984).Funding Heiko Dietrich: Supported by Australian Research Council grant DP190100317. Murray Elder: Supported by Australian Research Council grant DP210100271. Adam Piggott: Supported by Australian Research Council grant DP210100271. Youming Qiao: Supported by Australian Research Council grant DP200100950.enPublisher Copyright: © Heiko Dietrich, Murray Elder, Adam Piggott, Youming Qiao, and Armin Weiß.Inverse-closed finite convergent length-reducing rewriting systemIsomorphism problemPlain groupPolynomial hierarchyΣ complexity class2022-03-0110.4230/LIPIcs.STACS.2022.2685127101310