Measures in $$\mathcal {S}\mathcal {M}(\Gamma )$$ and Their Geometries

For any torsion-free finitely generated nilpotent group $$\Gamma $$ , this short but essential chapter introduces the set $$\mathcal {S}\mathcal {M}(\Gamma )$$ , a set of stable-like probability measures on $$\Gamma $$ . For each measure $$\mu $$ in $$\mathcal {S}\mathcal {M}(\Gamma )$$ , a particular “geometry” associated with $$\mu $$ is defined. This geometry will later be the key needed to understand how to define norms and appropriate approximate dilations adapted to the measure $$\mu $$ in order to apply the limit theorems of Chaps. 5 and 6 .