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.