Vesa Ala-Mattila: Covering measures, packing measures and Patterson-Sullivan measures of geometrically finite Kleinian groups

Abstract: I introduce new variants of covering and packing measure constructions and discuss them in the context of a non-elementary geometrically finite Kleinian group G.  More specifically, I discuss a theorem stating that these variants can be used to construct a measure v such that m=cv, where c>0 is a constant and m is a given Patterson-Sullivan measure of G.  This theorem is a generalization of a theorem proved by Dennis Sullivan which states that the measure v can sometimes be constructed using the standard covering construction and sometimes the standard packing construction; Sullivan proved also that sometimes neither of the two standard constructions can be used to construct v.