On groups presented by inverse-closed finite convergent length-reducing rewriting systems

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 triangles are uniformly bounded. This leads to a new algebraic result: the group is plain (isomorphic to the free product of finitely many finite groups and copies of $mathbb Z$) 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 that the isomorphism problem for plain groups presented by inverse-closed finite convergent length-reducing rewriting systems is in $mathsf{PSPACE}$.

Being Cayley automatic is closed under taking wreath product with virtually cyclic groups

We extend work of the first author and Khoussainov to show that being Cayley automatic is closed under taking the restricted wreath product with a virtually infinite cyclic group. This adds to the list of known examples of Cayley automatic groups.

Rewriting systems, plain groups, and geodetic graphs

We prove that a group is presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length 3 if and only if the group is plain. Our proof goes via a new result concerning properties of embedded circuits in geodetic graphs, which may be of independent interest in graph theory.

On the geometry of Cayley automatic groups

In contrast to being automatic, being Cayley automatic emph{a priori} has no geometric consequences. Specifically, Cayley graphs of automatic groups enjoy a fellow traveler property. Here we study a distance function introduced by the first author and Trakuldit which aims to measure how far a Cayley automatic group is from being automatic, in terms of how badly the Cayley graph fails the fellow traveler property. The first author and Trakuldit showed that if it fails by at most a constant amount, then the group is in fact automatic. In this article we show that for a large class of non-automatic Cayley automatic groups this function is bounded below by a linear function in a precise sense defined herein. In fact, for all Cayley automatic groups which have super-quadratic Dehn function, or which are not finitely presented, we can construct a non-decreasing function which (1) depends only on the group and (2) bounds from below the distance function for any Cayley automatic structure on the group.

A virtually 2-step nilpotent group with polynomial geodesic growth

A direct consequence of Gromovs theorem is that if a group has polynomial geodesic growth with respect to some finite generating set then it is virtually nilpotent. However, until now the only examples known were virtually abelian. In this note we furnish an example of a virtually 2-step nilpotent group having polynomial geodesic growth with respect to a certain finite generating set.

The complexity of solution sets to equations in hyperbolic groups

We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, as shortlex geodesic words (or any regular set of quasigeodesic normal forms), is an EDT0L language whose specification can be computed in NSPACE$(n^2log n)$ for the torsion-free case and NSPACE$(n^4log n)$ in the torsion case. Furthermore, in the presence of quasi-isometrically embeddable rational constraints, we show that the full set of solutions to systems of equations in a hyperbolic group remains EDT0L. Our work combines the geometric results of Rips, Sela, Dahmani and Guirardel on the decidability of the existential theory of hyperbolic groups with the work of computer scientists including Plandowski, Je.z, Diekert and others on PSPACE algorithms to solve equations in free monoids and groups using compression, and involves an intricate language-theoretic analysis.

$k$-pop stack sortable permutations and $2$-avoidance

We consider permutations sortable by $k$ passes through a deterministic pop stack. We show that for any $kinmathbb N$ the set is characterised by finitely many patterns, answering a question of Claesson and Gu{dh}mundsson. Our characterisation demands a more precise definition than in previous literature of what it means for a permutation to avoid a set of barred and unbarred patterns. We propose a new notion called emph{$2$-avoidance}.

Solutions sets to systems of equations in hyperbolic groups are EDT0L in PSPACE

We show that the full set of solutions to systems of equations and inequations in a hyperbolic group, with or without torsion, as shortlex geodesic words, is an EDT0L language whose specification can be computed in $mathsf{NSPACE}(n^2log n)$ for the torsion-free case and $mathsf{NSPACE}(n^4log n)$ in the torsion case. Our work combines deep geometric results by Rips, Sela, Dahmani and Guirardel on decidability of existential theories of hyperbolic groups, work of computer scientists including Plandowski, Je.z, Diekert and others on $mathsf{PSPACE}$ algorithms to solve equations in free monoids and groups using compression, and an intricate language-theoretic analysis. The present work gives an essentially optimal formal language description for all solutions in all hyperbolic groups, and an explicit and surprising low space complexity to compute them.

Permutations sorted by a finite and an infinite stack in series

We prove that the set of permutations sorted by a stack of depth $t geq 3$ and an infinite stack in series has infinite basis, by constructing an infinite antichain. This answers an open question on identifying the point at which, in a sorting process with two stacks in series, the basis changes from finite to infinite.

Solutions to twisted word equations and equations in virtually free groups

It is well known that the problem solving equations in virtually free groups can be reduced to the problem of solving twisted word equations with regular constraints over free monoids with involution. In this paper we prove that the set of all solutions of a twisted word equation is an EDT0L language whose specification can be computed in $mathsf{PSPACE}$. Within the same complexity bound we can decide whether the solution set is empty, finite, or infinite. In the second part of the paper we apply the results for twisted equations to obtain in $mathsf{PSPACE}$ an EDT0L description of the solution set of equations with rational constraints for finitely generated virtually free groups in standard normal forms with respect to a natural set of generators. If the rational constraints are given by a homomorphism into a fixed (or small enough) finite monoid, then our algorithms can be implemented in $mathsf{NSPACE}(n^2log n)$, that is, in quasi-quadratic nondeterministic space. Our results generalize the work by Lohrey and Senizergues (ICALP 2006) and Dahmani and Guirardel (J. of Topology 2010) with respect to both complexity and expressive power. Neither paper gave any concrete complexity bound and the results in these papers are stated for subsets of solutions only, whereas our results concern all solutions.