Infinite characters on GLn(Q), on SLn(Z), and on groups acting on trees

Abstract

Answering a question of J. Rosenberg from [21], we construct the first examples of infinite characters on GLn(K) for a global field K and n≥2. The case n=2 is deduced from the following more general result. Let G a non amenable countable group acting on a locally finite tree X. Assume either that the stabilizer in G of every vertex of X is finite or that the closure of the image of G in Aut(X) is not amenable. We show that G has uncountably many infinite dimensional irreducible unitary representations (π,H) of G which are traceable, that is, such that the C⁎-subalgebra of B(H) generated by π(G) contains the algebra of the compact operators on H. In the case n≥3, we prove the existence of infinitely many characters for G=GLn(R), when R is an integral domain such that G is not amenable. In particular, the group SLn(Z) has infinitely many such characters for n≥2.