Relative property (T) and linear groups

Abstract

Relative property (T) has recently been used to show the existence of a variety of new rigidity phenomena, for example in von Neumann algebras and the study of orbit-equivalence relations. However, until recently there were few examples of group pairs with relative property (T) available through the literature. This motivated the following result: A finitely generated group G admits a special linear representation with non-amenable R-Zariski closure if and only if it acts on an Abelian group A (of finite nonzero Q-rank) so that the corresponding group pair (G x A,A) has relative property (T). The proof is constructive. The main ingredients are Furstenberg’s celebrated lemma about invariant measures on projective spaces and the spectral theorem for the decomposition of unitary representations of Abelian groups. Methods from algebraic group theory, such as the restriction of scalars functor, are also employed.