Valette, Alain

2008, Bekka, Bachir, Valette, Alain, De La Harpe, Pierre

1993, De La Harpe, Pierre, Robertson, Guyan, Valette, Alain

Let GAMMA be a finitely generated group. In the group algebra C[T], form the average h of a finite set S of generators of GAMMA. Given a unitary representation pi of GAMMA, we relate spectral properties of the operator pi(h) to ProPerties Of GAMMA and pi. For the universal representation pi(un) of GAMMA, we prove in particular the following results. First, the spectrum Sp(pi(un) (h)) contains the complex number , of modulus one iff Sp(pi(un) (h)) is invariant under multiplication by z, iff there exists a character x: GAMMA --> T such that chi(S) = {z}. Second, for S-1 = S, the group GAMMA has Kazhdan's proPertY (T) if and only if 1 is isolated in Sp(pi(un) (h)); in this case, the distance between 1 and other point, of the spectrum gives a lower bound on the Kazhdan constants. Numerous examples illustrate the results.

1994, De La Harpe, Pierre, Robertson, Guyan, Valette, Alain

1989, De La Harpe, Pierre, Valette, Alain

1993, De La Harpe, Pierre, Robertson, Guyan, Valette, Alain