Free nilpotent groups are -superrigid

The free nilpotent group G m , n G_{m,n} of class m m and rank n n is the free object on n n generators in the category of nilpotent groups of class at most m m . We show that G m , n G_{m,n} can be recovered from its reduced group C ∗ C^* -algebra, in the sense that if H H is any group such that C r ∗ ( H ) C^*_r(H) is isomorphic to C r ∗ ( G m , n ) C^*_r(G_{m,n}) , then H H must be isomorphic to G m , n G_{m,n} .