Quasi-isometric classification of non-geometric 3-manifold groups (Q2910996)

The authors study the quasi-isometric classification of certain non-geometric \(3\)-manifold groups. They concentrate on the case where all of the geometric pieces are hyperbolic and at least one of which is non-arithmetic, though they extend to the case where Seifert fibered pieces are allowed.NEWLINENEWLINEIn order to carry out the classification, the authors associate to a manifold in the class under consideration a certain labeled graph. There is an equivalence relation on these labeled graphs, called bisimilarity.NEWLINENEWLINEThe authors prove that the bisimilarity class of the labeled graph determines the quasi-isometry type of the manifold to which the graph was associated. They show that every labeled graph (satisfying some minimality condition) is the labeled graph associated to a quasi-isometry class of manifolds under consideration. Finally, they give a necessary condition for two manifolds in the class under consideration to be commensurable.