Separable Universal Ii

Gromov constructed uncountably many pairwise non-isomorphic discrete groups with Kazhdan's property (T). We will show that no separable II 1-factor can contain all these groups in its unitary group. In particular, no separable II 1-factor can contain all separable II 1-factors in it. We also show that the full group C *-algebras of some of these groups fail the lifting property. We recall that a discrete group Γ is said to have Kazhdan's property (T) if the trivial representation is isolated in the duaî Γ of Γ, equipped with the Fell topology. This is equivalent to that there are a finite subset E of generators in Γ and a decreasing function f : R + → R + with lim ε→0 f (ε) = 0 such that the following is true: if π is a unitary representation of Γ on a Hilbert space H and ξ ∈ H is a unit vector with ε = max s∈E π(s)ξ − ξ, then there is a vector η ∈ H with ξ−η < f (ε) (in particular η = 0 when ε is small enough) such that π(s)η = η for all s ∈ Γ. We refer the reader to [HV] and [V] for the information of Kazhdan's property (T). We recall that a discrete group Γ is said to be quasifinite if all its proper subgroups are finite, and is said to be infinite conjugacy classes (abbreviated to ICC) if all nontrivial conjugacy classes in Γ are infinite. We note that a discrete group Γ is ICC if and only if its group von Neumann algebra LΓ is a factor. We also observe that a group which is quasifinite and ICC has to be simple. Gromov (Corollary 5.5.E in [G]) claimed that any torsion-free non-cyclic hy-perbolic group has a quotient group all of whose proper subgroups are cyclic of prescribed orders (cf. Theorem 3.4 in [V]). This claim was partly confirmed by Olshanskii (Corollary 4 in [O]). Actually, what Olshanskii proved there is that any torsion-free non-cyclic hyperbolic group has a nontrivial quasifinite quotient group. We observe that Olshanskii's argument gives us the following. Theorem 1 (Gromov-Olshanskii). Any torsion-free non-cyclic hyperbolic group has uncountably many pairwise non-isomorphic quotient groups all of which are quasifi-nite and ICC. In particular, there is a discrete group Γ with Kazhdan's property (T) which has uncountably many pairwise non-isomorphic quotient groups {Γ α } α∈I all …