Nearly-linear solution to the word problem for 3-manifold groups

We show that the word problem for any 3-manifold group is solvable in time $O(n{\log }^{3}n)$. Our main contribution is the proof that the word problem for admissible graphs of groups, in the sense of Croke and Kleiner, is solvable in $O(n\log n)$; this covers fundamental groups of non-geometric graph manifolds. Similar methods also give that the word problem for free products can be solved ``almost as quickly’’ as the word problem in the factors.