We consider crossed product II^sub 1^ factors (ProQuest: Formulae and/or non-USASCII text omitted; see image) , with G discrete ICC groups that contain infinite normal subgroups with the relative property (T) and σ trace preserving actions of G on finite von Neumann algebras N that are "malleable" and mixing. Examples are the actions of G by Bernoulli shifts (classical and non-classical) and by Bogoliubov shifts. We prove a rigidity result for isomorphisms of such factors, showing the uniqueness, up to unitary conjugacy, of the position of the group von Neumann algebra L(G) inside M. We use this result to calculate the fundamental group of M, (ProQuest: Formulae and/or non-USASCII text omitted; see image) , in terms of the weights of the shift σ, for (ProQuest: Formulae and/or non-USASCII text omitted; see image) and other special arithmetic groups. We deduce that for any subgroup S^sub +^^sup *^ there exist II^sub 1^ factors M (separable if S is countable or S=^sub +^^sup *^) with (ProQuest: Formulae and/or non-USASCII text omitted; see image) . This brings new light to a long standing open problem of Murray and von Neumann. PUBLICATION ABSTRACT