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 of a compact 3–manifold, a finite generating set for a subgroup , and an element , and determines whether or not .
"The membership problem for $3$–manifold groups is solvable." Algebr. Geom. Topol. 16 (4) 1827 - 1850, 2016. https://doi.org/10.2140/agt.2016.16.1827