Almost representations and asymptotic representations of discrete groups

We define for discrete finitely presented groups a new property related to their asymptotic representations. Namely we say that a groups has the property AGA if every almost representation generates an asymptotic representation. We give examples of groups with and without this property. For our example of a group Γ without AGA the group K(BΓ) cannot be covered by asymptotic representations of Γ. One of the reasons of attention to almost and asymptotic representations of discrete groups [5, 2] is their relation to K-theory of classifying spaces [2, 9]. It was shown in [9] that in the case of finite-dimensional classifying space BΓ in order to construct a vector bundle over it out of an asymptotic representation of Γ it is sufficient to have an ε-almost representation of Γ with small enough ε. Of course an ε-almost representation contains less information than the whole asymptotic representation, but it turns out that often the information contained in an ε-almost representation makes it possible to construct the corresponding asymptotic representation. In the present paper we give the definition of this property, prove this property for some classes of groups and finally give an example of a group without this property. We discuss also this example in relation with its K-theory. 1 Basic definitions Let Γ be a finitely presented discrete group, and let Γ = 〈F |R〉 = 〈g1, . . . , gn|r1, . . . , rk〉 be its presentation with gi being generators and rj = rj(g1, . . . , gn) being relations. We assume that the set F = {g1, . . . , gn} is symmetric, i.e. for every gi it contains g −1 i too, and the set R of relations contains relations of the form gig −1 i , though we usually will skip these additional generators and relations. By U∞ we denote the direct limit of the groups Un with respect to the natural inclusion Un−→Un+1 supplied with the standard operator norm. The unit matrix we denote by I ∈ U∞. Definition 1 A set of unitaries σ = {u1, . . . , un} ⊂ U∞ is called an ε-almost representation of the group Γ if after substitution of ui istead of gi, i = 1, . . . , n, into rj one has ‖rj(u1, . . . , un)− I‖ ≤ ε for all j = 1, . . . , k. 1 In this case we write σ(gi) = ui. Remark that this definition depends on a choice of presentation of the group Γ, but we will see that this dependence is not important. Let 〈h1, . . . , hm|s1, . . . , sl〉 be another presentation of Γ. For an ε-almost representation σ with respect to the first presentation we can define the set of unitaries v1 . . . , vm ∈ U∞, vi = σ(hi) putting σ(hi) = σ(gj1) · . . . ·σ(gjni ), where hi = gj1 · . . . · gjni . By the same way starting from the set σ(hi) we can construct the set σ(gi). Lemma 2 There exist constants C and D (depending on the two presentations) such that σ is a Cε-almost representation with respect to the second presentation of Γ and for all gi, i = 1, . . . , n, one has ‖σ(gi)− σ(gi)‖ ≤ Dε. Proof. We have to estimate the norms ‖sq(v1, . . . , vm)− I‖, q = 1, . . . , l. To do so notice that every relation sq can be written in the form sq = a −1 1 r ǫ1 j1a1 · . . . · a −1 mqr ǫmq jmq amq (1) for some ai ∈ Γ, where ǫi = ±1. Let M be the maximal length of the words ai = ai(g1, . . . , gn). Put b ′ i = a −1 i (g1, . . . , gn) ∈ U∞, bi = ai(g1, . . . , gn) ∈ U∞. Then one has ‖b′ibi − I‖ ≤ Mε. It follows from (1) that sq(v1, . . . , vm) = b ′ 1rj1(u1, . . . , un)b1 · . . . · b ′ mqrjmq (u1, . . . , un)bmq , but as for every i one has