Finding tracks in 2-complexes

Abstract: I will present an algorithm which given as input --- (a) a
finite presentation < X | R > for a group without 2-torsion, (b) a
solution to the word problem with respect to this presentation and (c)
a positive integer k --- outputs a finite collection t1,...,tn of
tracks with the property that if the group < X | R > admits a
k-acylindrical geometric splitting, then up to automorphism, an edge
group of this splitting is carried by one of these tracks.

I will define what all this means and give some applications, namely
the detection of non-trivial free decompositions and the detection of
parabolic splittings of relatively hyperbolic groups.