Detecting geometric splittings in finitely presented groups

We present an algorithm which given a presentation of a group G without 2-torsion, a solution to the word problem with respect to this presentation, and an acylindricity constant \kappa outputs a collection of tracks in an appropriate presentation complex. We give two applications: the first is an algorithm which decides if G admits an essential free decomposition; the second is an algorithm which, if G is relatively hyperbolic, decides if it admits an essential elementary splitting.