# Decision problems for 3-manifolds and their fundamental groups

@article{Aschenbrenner2014DecisionPF, title={Decision problems for 3-manifolds and their fundamental groups}, author={Matthias Aschenbrenner and Stefan Friedl and Henry Wilton}, journal={arXiv: Geometric Topology}, year={2014} }

We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (Word Problem, Conjugacy Problem, Isomorphism Problem), and also the Homeomorphism Problem for 3-manifolds and the Membership Problem for 3-manifold groups.

## 31 Citations

### The membership problem for 3–manifold groups is solvable

- Mathematics
- 2016

We show that the membership problem for finitely generated subgroups of 3-manifold groups is uniformly solvable. That is, there is an algorithm that takes as input a presentation for the fundamental…

### Conjugacy Problem in the Fundamental Groups of High-Dimensional Graph Manifolds

- Mathematics, Computer ScienceMathematics
- 2021

The graph of group structure for the fundamental group of a high-dimensional graph manifold and study the conjugacy problem is analyzed and a new proof for the solvable word problem is provided.

### The Conjugacy Problem for ascending HNN-extensions of free groups

- Mathematics
- 2022

We give an algorithm to solve the Conjugacy Problem for ascending HNN-extensions of free groups. To do this, we give algorithms to solve certain problems on dynamics of free group

### 2-manifold Recognition Is in Logspace

- MathematicsJ. Comput. Geom.
- 2016

It is proved that the homeomorphism problem for 2-manifolds can be decided in logspace using Reingold's logspace solution to the undirected $s,t$-connectivity problem in graphs.

### 2-Stratifold groups have solvable word problem

- Mathematics
- 2017

Abstract2-Stratifolds are a generalization of 2-manifolds in that there are disjoint simple closed curves where several sheets meet. We show that the word problem for fundamental groups of…

### 3-Manifold Groups

- Mathematics
- 2015

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

### Toward and after virtual specialization in $3$-manifold topology

- MathematicsSurveys in Differential Geometry
- 2020

. This article surveys works in 3 –manifold topology after the proof of the virtual Haken conjecture. It reviews the virtual specialization of 3 –manifold groups, and some following development of…

### 2-stratifold spines of closed 3-manifolds

- Mathematics
- 2017

$2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed branch curves. We obtain a list of all closed $3$-manifolds that have a $2$-stratifold as a spine.

### AN INTRODUCTION TO 3-MANIFOLDS AND THEIR FUNDAMENTAL GROUPS

- Mathematics
- 2015

In these lecture notes we will give a quick introduction to 3-manifolds, with a special emphasis on their fundamental groups. In the first section we will show that given k ≥ 4 any finitely presented…

### All Dehn Fillings of the Whitehead Link Complement are Tetrahedron Manifolds

- MathematicsInternational Electronic Journal of Geometry
- 2022

In this paper we show that Dehn surgeries on the oriented components of the Whitehead link yield tetrahedron manifolds of Heegaard genus $\le 2$. As a consequence, the eight homogeneous Thurston…

## References

SHOWING 1-10 OF 141 REFERENCES

### The membership problem for 3–manifold groups is solvable

- Mathematics
- 2016

We show that the membership problem for finitely generated subgroups of 3-manifold groups is uniformly solvable. That is, there is an algorithm that takes as input a presentation for the fundamental…

### An Algorithm to Recognize the 3-Sphere

- Mathematics
- 1995

Max Dehn introduced a number of important decision problems in algebra and topology. For groups, the isomorphism problem is to decide if two finite presentations represent isomorphic groups or not.…

### Recognizing geometric 3-manifold groups using the word problem

- Mathematics
- 2012

Adyan and Rabin showed that most properties of groups cannot be algorithmically recognized from a finite presentation alone. We prove that, if one is also given a solution to the word problem, then…

### The conjugacy problem in groups of non-orientable 3-manifolds

- Mathematics
- 2012

We prove that fundamental groups of non-orientable 3-manifolds have a solvable conjugacy problem, and construct an algorithm. Together with our earlier work on the conjugacy problem in groups on…

### The conjugacy problem in groups of orientable geometrizable 3-manifolds

- Mathematics
- 2013

We prove that fundamental groups of orientable (geometrizable) 3-manifolds have a solvable conjugacy problem.

### Algorithmic Topology and Classification of 3-Manifolds

- Mathematics
- 2003

Simple and special polyhedra.- Complexity theory of 3-manifolds.- Haken theory of normal surfaces.- Applications of the theory of normal surfaces.- Algorithmic recognition of S3.- Classification of…

### The homeomorphism problem for closed 3–manifolds

- Mathematics
- 2014

We give a geometric approach to an algorithm for deciding whether two hyperbolic 3‐manifolds are homeomorphic. We also give an algebraic approach to the homeomorphism problem for geometric, but…

### Shrinkwrapping and the taming of hyperbolic 3-manifolds : preprint

- Mathematics
- 2004

We introduce a new technique for finding CAT(-1) surfaces in hyperbolic 3-manifolds. We use this to show that a complete hyperbolic 3-manifold with finitely generated fundamental group is…

### Tameness of hyperbolic 3-manifolds

- Mathematics
- 2004

We show that hyperbolic 3-manifolds with finitely generated fundamental group are tame, that is the ends are products. We actually work in slightly greater generality with pinched negatively curved…

### On Kneser's conjecture for bounded 3-manifolds

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1972

1. The classical Kneser conjecture states that if M is a closed 3-manifold and π1(M) ≈ A * B, a free product, then there exist 3-manifolds MA, MB such that and π1(MA) ≈ A, π1(MB) ≈ B. This has been…