# Enumerating limit groups

@article{Groves2007EnumeratingLG, title={Enumerating limit groups}, author={Daniel Groves and Henry Wilton}, journal={Groups, Geometry, and Dynamics}, year={2007}, volume={3}, pages={389-399} }

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…

## 18 Citations

### On the difficulty of presenting finitely presentable groups

- Mathematics
- 2011

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

- Mathematics
- 2009

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

- MathematicsInt. J. Algebra Comput.
- 2013

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

- Mathematics
- 2016

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

- Mathematics
- 2010

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

- MathematicsTransactions 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

- Mathematics
- 2016

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

- MathematicsInt. J. Algebra Comput.
- 2013

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

- Mathematics
- 2013

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

- Mathematics
- 2008

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…

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