Decision problems for 3-manifolds and their fundamental groups

TL;DR

This paper surveys the current state of decision problems related to 3-manifolds and their fundamental groups, covering classical problems for groups and specific topological problems for 3-manifolds.

Contribution

It provides a comprehensive overview of the status and progress on key decision problems in 3-manifold topology and group theory.

Findings

Classical decision problems for finitely presented groups are discussed.

Status of the Homeomorphism Problem for 3-manifolds is summarized.

The Membership Problem for 3-manifold groups is included.

Abstract

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.