Groups with presentations in EDT0L

Abstract

To any family of languages LAN, let us associate the class, denoted $\pi (\mathrm{LAN})$, of finitely generated groups that admit a group presentation whose set of relators forms a language in LAN. We show that the class of finitely L-presented groups, obtained by iterating finitely many endomorphisms on finitely many relations, is exactly the class of groups that admit a presentation in the family of languages EDT0L. We show that the marked isomorphism problem is not semi-decidable for groups given by EDT0L presentations, contrary to the finite presentation case. We then extend and unify results of the first author with Eick and Hartung about nilpotent and finite quotients, by showing that it is possible to compute the marked hyperbolic and marked abelian-by-nilpotent quotients of a group given by an EDT0L presentation. Finally, we show how the results about quotient computations enable the construction of recursively presented groups that do not have EDT0L presentations, thus proving $\pi (EDT0L)\ne \pi (\mathrm{REC})$. This is done by building a residually nilpotent group with solvable word problem whose sequence of maximal nilpotent quotients is non-computable.