No Separable Ii1-factor Can Contain All Separable Ii1-factors as Its Subfactors

Gromov gave an uncountable family of countable discrete groups with Kazhdan’s property (T). In this note, we will show that there is no separable II1-factor whose unitary group contains all these groups of Gromov as its subgroups. In particular, there is no separable II1-factor which contains all separable II1-factors as its subfactors. Recall that a discrete group Γ is said to have Kazhdan’s property (T) if the trivial representation is isolated in the dual Γ̂ of Γ, equipped with the Fell topology. This is equivalent to that there are a finite subset E of generators in Γ, a constant κ > 0 and a decreasing function f : R+ → R+ with limε→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 = maxs∈E ‖π(s)ξ− ξ‖ < κ, then there is a unit vector η ∈ H with ‖ξ − η‖ < f(ε) such that π(s)η = η for all s ∈ Γ. We refer the reader to [2] and [7] for the information of Kazhdan’s property (T). Gromov (Corollary 5.5.E in [1]) proved that there is a countable discrete group Γ with Kazhdan’s property (T) which has uncountably many pairwise non-isomorphic quotients {Γi}i∈I which are simple and ICC. Simplicity of these groups was noticed by Cherix. See Theorem 3.4 and the following remarks in [7]. Connes conjectured that a discrete group Λ with Kazhdan’s property (T) and ICC property is uniquely determined by its group von Neumann algebra LΛ. The following theorem, in particular, confirms Connes’ conjecture for {Γi}i∈I “modulo countable sets”. Theorem 1. Let {Γi}i∈I be as above. If M is a separable II1-factor, then the set {i ∈ I : the unitary group U(M) of M contains a subgroup isomorphic to Γi} is at most countable. The following corollary, which was suggested by S. Popa, generalizes McDuff’s theorem [3] and solves Problem 4.4.29 of [6] and Conjecture 4.5.5 in [4] negatively. See also Theorem 1 in [5] and its remarks. Recall that two II1-factors M and N are said to be stable equivalent if there are n ∈ N and a projection p ∈ Mn(M) such that pMn(M)p is isomorphic to N . Corollary 2. Let {Γi}i∈I be as above. If M is a separable II1-factor, then the set {i ∈ I : M contains a subfactor which is stable equivalent to LΓi} is at most countable. 1991 Mathematics Subject Classification. Primary 46L10; Secondary 20F65.