Star-complexes, and the dependence problems for hyperbolic complexes

Abstract

Given a group presentation (or more generally† a 2-complex) one can associate with it an object which has variously been called the co-initial graph, star-graph, star-complex, and which has proved useful in several contexts [2], [6], [7], [8], [9], [10], [12]. For certain mappings of 2-complexes φ: ⃗ℒ (”strong mappings”) one gets an induced mapping φst: st⃗ℒst of the associated star-complexes. Then st is a covariant functor from the category of 2-complexes (where the morphisms are strong mappings) to the category of 1-complexes, and this functor behaves very nicely with respect to coverings (Theorem 1).