Property T of group homomorphisms

In this article, we define and study property T for continuous homomorphisms between topological groups. If G   is a locally compact group, we show that λG:G→U(L2(G))λG:G→U(L2(G)) has property T if and only if either G is compact or G   is non-amenable. Moreover, the abelianization Gab:=G/[G,G]‾ is compact if and only if every continuous homomorphism from G to any abelian topological group has property T. Moreover, we show that G   has property (T,FD)(T,FD) if and only if any continuous homomorphism from G to any compact group has property T. In the case when G is almost connected, the above is also equivalent to the canonical map from G to its Bohr compactification being a quotient map. We also give some new equivalent forms of the strong property T of a locally compact group. As a consequence, if G is a second countable and has strong property T and H is a closed subgroup of G, there exist at most one G  -invariant mean on G/HG/H.