Infinite characters on GL(Q), on SL (Z), and on groups acting on trees

• Published in: Journal of functional analysis Vol. 277; no. 7; pp. 2160 - 2178
• Main Author: Bekka, Bachir
• Format: Journal Article
• Language: English
• Published: Elsevier 01.10.2019
• Subjects: Functional Analysis; Mathematics
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2018.10.003

Answering a question of J. Rosenberg from [Ros–89], we construct the first examples of infinite characters on GL n (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 subgroup acting on 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 = GL n (R), where n ≥ 3 and R is an integral domain such that G is not amenable. In particular, the group SL n (Z) has infinitely many such characters for n ≥ 2.