Cluster categories and their relation to Cluster algebras, Semi-invariants and Homology of torsion free nilpotent groups

the structure of cluster categories [bmrrt] is well suited for the combinatorics of cluster algebras [fz1] with the main correspondence being between tilting objects and clusters. furthermore it was shown in [iotw] that there is a close relation between domains of virtual semi-invariants and simplicial complexes associated to cluster categories. also the same simplicial complexes associated to cluster categories are related to the igusa-orr pictures in the homology of nilpotent groups.