The word problem in a class of non-Haken 3-manifolds (Q1327325)

Let \(M\) be a closed \(P^2\)-irreducible, non-Haken 3-manifold that admits a singular incompressible surface (different from a singular \(S^2\) or \(P^2\)). Hass, Rubinstein, and Scott showed that then \(M\) admits a singular incompressible surface \(F\) whose lifts to the universal cover of \(M\) are embedded planes any two of which intersect transversely in a collection of lines. The main result of the paper is that the word problem for \(\pi_1 (M)\) is solvable, provided that the lifts of \(F\) form a family \(\Pi\) satisfying the following properties: any two planes from \(\Pi\) are either disjoint or intersect in a single line and any three planes from \(\Pi\) have at most one point in common. This result is then applied to solve the word problem for \(\pi_1 (M)\) in a class of closed 3-manifolds \(M\) which were not previously known to have solvable word problems.