Conjugacy problem in groups of oriented geometrizable 3-manifolds (Q2570065)

The aim of the paper is to show that the fundamental group of an oriented 3-manifold which satisfies Thurston's geometrization conjecture has a solvable conjugacy problem. In other words, for any such 3-manifold \(M\) there exists an algorithm which can decide for any pair of elements \(u,v\in\pi_1(M)\) whether \(u\) and \(v\) are in the same conjugacy class or not. More topologically, the algorithm decides for any two loops in \(M\) whether they are freely homotopic. A 3-manifold is said to satisfy Thurston's geometrization conjecture if it decomposes in its canonical decomposition along essential spheres, disks and tori into pieces whose interiors admit a complete and locally homogeneous Riemannian metric.