Asymptotic Representation of Discrete Groups

Abstract

Abstract.If one has a unitary representation p : 7r -+ U(H) of the fundamental group 7rl(M) of the manifold M, then one can do many useful things: 1) construct a natural vector bundle over M; 2) construct the cohomology groups with respect to the local system of coefficients; 3) construct the signature of manifold M with respect to the local system of coefficients; and others. In particular, one can write the Hirzebruch formula, which compares the signature with the characteristic classes of the manifold M, further, based on this, find the homotopy invariant characteristic classes (i.e., the Novikov conjecture). Taking into account that the family of known representations is not sufficiently large, it would be interesting to extend this family to some larger one. Using the ideas of P. de la Harpe and M. Karoubi

Almost-representations and asymptotic representations of discrete groups

“…To get such representation we can take σ(b) to be the infinite direct sum of matrices of the form 0 1 1 0 and to rearrange the basis for the matrices σ n (a) and σ n (c) in such a way that σ n (a) and σ(b) would almost commute. We should also insert a number of intermediate matrices between σ n and σ n+1 as it was done in [9]. Denote by H n the Hilbert space where the matrices σ n (a), σ n (c) and σ(b) act and put H = ⊕ n H n (for negative n put σ n (a) = σ n (c) = I).…”

“…Remember that in [9] a map R asym (Γ)−→K 0 (BΓ) (10) was constructed, which factorizes [8] through the assembly map…”

“…We give examples of groups with and without this property. For our example of a group Γ without AGA the group K 0 (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.…”

Vector bundles from generalized pairs of cocycles

“…It is known that the cocycle condition can be weakened and still results in locally trivial bundles. It was shown by Mishchenko in [3] that (1.1) can be replaced by g αβ g βγ − g αγ < ε with ε sufficiently small, and these ε-cocycles still determine vector bundles. Locally flat bundles are often determined by representations π 1 (X) → U(N ), and Mishchenko's construction obtains vector bundles from the so-called almost representations.…”

Проблема Каждана - Мильмана Для Полупростых Компактных Групп Ли

“…также [29], [102], [125], [126]). В них речь идет об отображениях, удовлетворяющих условию ограниченности или малости не на всей группе, а на некотором компакте (в частности, на множе-стве образующих конечно порожденной дискретной группы).…”