Plain groups and rewriting systems.

In recent work joint with Adam Piggott (ANU), we show that groups presented by inverse-closed finite convergent length-reducing rewriting systems are characterised by a striking geometric property: their Cayley graphs are geodetic and side-lengths of non-degenerate geodesic triangles are uniformly bounded. This leads to a new algebraic result: the group is plain (isomorphic to a free product of finitely many finite groups and a finite-rank free group) if and only if a certain relation on the set of non-trivial finite-order elements of the group is transitive on a bounded set. We use this to prove that deciding if a group presented by an inverse-closed finite convergent length-reducing rewriting system is not plain is in $\mathsf{NP}$. A “yes” answer would disprove a longstanding conjecture of Madlener and Otto from 1987.
We also prove (joint with Dietrich, Piggott, Qiao and Weiß) that the isomorphism problem for plain groups presented by inverse-closed finite convergent length-reducing rewriting systems lies in the polynomial time hierarchy, more precisely, in the complexity class $\Sigma_3^{\mathsf{P}}$.
The talk is based on the papers https://arxiv.org/abs/2106.03445 and https://arxiv.org/abs/2110.00900