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

Abstract

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 group $\pi$ of a compact 3–manifold, a finite generating set for a subgroup $\Gamma$, and an element $g\in \pi$, and determines whether or not $g\in \Gamma$.