On the difficulty of presenting finitely presentable groups

We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of integer matrices, and right-angled Coxeter groups form such classes. We discuss related classes of groups in which there does exist an algorithm to compute finite presentations for finitely presentable subgroups. We also construct a finitely presented group that has a polynomial Dehn function but in which there is no algorithm to compute the first Betti number of the finitely presentable subgroups.Comment: Final version. To appear in GGD volume dedicated to Fritz Grunewal

Stallings graphs for quasi-convex subgroups

We show that one can define and effectively compute Stallings graphs for quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or right-angled Artin groups). These Stallings graphs are finite labeled graphs, which are canonically associated with the corresponding subgroups. We show that this notion of Stallings graphs allows a unified approach to many algorithmic problems: some which had already been solved like the generalized membership problem or the computation of a quasi-convexity constant (Kapovich, 1996); and others such as the computation of intersections, the conjugacy or the almost malnormality problems. Our results extend earlier algorithmic results for the more restricted class of virtually free groups. We also extend our construction to relatively quasi-convex subgroups of relatively hyperbolic groups, under certain additional conditions.Comment: 40 pages. New and improved versio

String rewriting for Double Coset Systems

In this paper we show how string rewriting methods can be applied to give a new method of computing double cosets. Previous methods for double cosets were enumerative and thus restricted to finite examples. Our rewriting methods do not suffer this restriction and we present some examples of infinite double coset systems which can now easily be solved using our approach. Even when both enumerative and rewriting techniques are present, our rewriting methods will be competitive because they i) do not require the preliminary calculation of cosets; and ii) as with single coset problems, there are many examples for which rewriting is more effective than enumeration. Automata provide the means for identifying expressions for normal forms in infinite situations and we show how they may be constructed in this setting. Further, related results on logged string rewriting for monoid presentations are exploited to show how witnesses for the computations can be provided and how information about the subgroups and the relations between them can be extracted. Finally, we discuss how the double coset problem is a special case of the problem of computing induced actions of categories which demonstrates that our rewriting methods are applicable to a much wider class of problems than just the double coset problem.Comment: accepted for publication by the Journal of Symbolic Computatio

Deciding Isomorphy using Dehn fillings, the splitting case

We solve Dehn's isomorphism problem for virtually torsion-free relatively hyperbolic groups with nilpotent parabolic subgroups. We do so by reducing the isomorphism problem to three algorithmic problems in the parabolic subgroups, namely the isomorphism problem, separation of torsion (in their outer automorphism groups) by congruences, and the mixed Whitehead problem, an automorphism group orbit problem. The first step of the reduction is to compute canonical JSJ decompositions. Dehn fillings and the given solutions of the algorithmic problems in the parabolic groups are then used to decide if the graphs of groups have isomorphic vertex groups and, if so, whether a global isomorphism can be assembled. For the class of finitely generated nilpotent groups, we give solutions to these algorithmic problems by using the arithmetic nature of these groups and of their automorphism groups.Comment: 76 pages. This version incorporates referee comments and corrections. The main changes to the previous version are a better treatment of the algorithmic recognition and presentation of virtually cyclic subgroups and a new proof of a rigidity criterion obtained by passing to a torsion-free finite index subgroup. The previous proof relied on an incorrect result. To appear in Inventiones Mathematica

Kleinian groups and the rank problem

We prove that the rank problem is decidable in the class of torsion-free word-hyperbolic Kleinian groups. We also show that every group in this class has only finitely many Nielsen equivalence classes of generating sets of a given cardinality.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper12.abs.htm

Orbit decidability and the conjugacy problem for some extensions of groups

Given a short exact sequence of groups with certain conditions, 1 ? F ? G ? H ? 1, weprove that G has solvable conjugacy problem if and only if the corresponding action subgroupA 6 Aut(F) is orbit decidable. From this, we deduce that the conjugacy problem is solvable,among others, for all groups of the form Z2?Fm, F2?Fm, Fn?Z, and Zn?A Fm with virtually solvable action group A 6 GLn(Z). Also, we give an easy way of constructing groups of the form Z4?Fn and F3?Fn with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in Aut(F2) is given

Orbit decidability and the conjugacy problem for some extensions of groups

Given a short exact sequence of groups with certain conditions, $1\to F\to G\to H\to 1$, we prove that $G$ has solvable conjugacy problem if and only if the corresponding action subgroup $A\leqslant Aut(F)$ is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form $\mathbb{Z}^2\rtimes F_m$, $F_2\rtimes F_m$, $F_n \rtimes \mathbb{Z}$, and $\mathbb{Z}^n \rtimes_A F_m$ with virtually solvable action group $A\leqslant GL_n(\mathbb{Z})$. Also, we give an easy way of constructing groups of the form $\mathbb{Z}^4\rtimes F_n$ and $F_3\rtimes F_n$ with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in $Aut(F_2)$ is given