Enumerating limit groups

  title={Enumerating limit groups},
  author={Daniel Groves and Henry Wilton},
  journal={Groups, Geometry, and Dynamics},
We prove that the set of limit groups is recursively enumerable, answering a question of Delzant. One ingredient of the proof is the observation that a finitely presented group with local retractions (a la Long and Reid) is coherent and, furthermore, there exists an algorithm that computes presentations for finitely generated subgroups. The other main ingredient is the ability to algorithmically calculate centralizers in relatively hyperbolic groups. Applications include the existence of… 

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

Computing equations for residually free groups

We show that there is no algorithm deciding whether the maximal residually free quotient of a given finitely presented group is finitely presentable or not. Given a finitely generated subgroup G of a

The Subgroup Identification Problem for Finitely Presented Groups

The subgroup identification problem is introduced, there is a finitely presented group G for which it is unsolvable, and it is uniformly solvable in the class offinitely presented locally Hopfian groups as an investigation into the difference between strong and weak effective coherence for finitelyPresent groups.

Detecting geometric splittings in finitely presented groups

We present an algorithm which given a presentation of a group $G$ without 2-torsion, a solution to the word problem with respect to this presentation, and an acylindricity constant ${\kappa}$,

Conjugacy classes of solutions to equations and inequations over hyperbolic groups

We study conjugacy classes of solutions to systems of equations and inequations over torsion‐free hyperbolic groups, and describe an algorithm to recognize whether or not there are finitely many

Homomorphisms to acylindrically hyperbolic groups I: Equationally noetherian groups and families

  • D. GrovesM. Hull
  • Mathematics
    Transactions of the American Mathematical Society
  • 2019
We study the set of homomorphisms from a fixed finitely generated group G G into a family of groups G \mathcal {G} which are ‘uniformly acylindrically hyperbolic’. Our main

The structure of limit groups over hyperbolic groups

Let Γ be a torsion-free hyperbolic group. We study Γ-limit groups which, unlike the fundamental case in which Γ is free, may not be finitely presentable or geometrically tractable. We define model

Actions, Length Functions, and Non-Archimedean Words

This paper surveys recent developments in the theory of groups acting on Λ-trees in an attempt to present various faces of the theory and to show how these methods can be used to solve major problems about finitely presented Κ-free groups.

Effective coherence of groups discriminated by a locally quasi-convex hyperbolic group

We prove that every finitely generated group $G$ discriminated by a locally quasi-convex torsion-free hyperbolic group $\Gamma$ is effectively coherent: that is, presentations for finitely generated

Finitely presented residually free groups

We establish a general criterion for the finite presentability of subdirect products of groups and use this to characterize finitely presented residually free groups. We prove that, for all



Hall’s Theorem for Limit Groups

Abstract.A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index

Combination of convergence groups

We state and prove a combination theorem for relatively hyperbolic groups seen as geometrically finite convergence groups. For that, we explain how to contruct a boundary for a group that is an

Existential questions in (relatively) hyperbolic groups

We study the decidability of the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the

Diophantine geometry over groups VII: The elementary theory of a hyperbolic group

This paper generalizes our work on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free

Finding relative hyperbolic structures

We propose an algorithm that recognizes relatively hyperbolic groups from a compatible relative presentation, and even from an arbitrary finite presentation when the parabolic subgroups are abelian.

Diophanting geometry over groups II: Completions, closures and formal solutions

This paper is the second in a series on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free

Irreducible Affine Varieties over a Free Group: II. Systems in Triangular Quasi-quadratic Form and Description of Residually Free Groups

Abstract We shall prove the conjecture of Myasnikov and Remeslennikov [ 4 ] which states that a finitely generated group is fully residually free (every finite set of nontrivial elements has

A Combination Theorem for Relatively Hyperbolic Groups

Given a graph of δ‐hyperbolic spaces, this paper gives sufficient conditions that ensure that the graph itself is δ‐hyperbolic. As an application, a simple proof is given to show that limit groups