On Kazhdan’s property (T) for the special linear group of holomorphic functions. (English) Zbl 1315.22006

Ivarsson, Björn; Kutzschebauch, Frank

Bull. Belg. Math. Soc. - Simon Stevin 21, No. 1, 185-191 (2014).

The main result of the paper states that if \(n\geq 3\) and \(X\) is a Stein manifold having finitely many connected components for which all holomorphic maps from it to SL\(_{n}(\mathbb C)\) are null-homotopic, then SL\(_{n}(\mathcal O (X))\) has Kazhdan’s property (T). This property, which expresses a certain rigidity for unitary Hilbert spaces representations of a topological group is presented in the paper. Several interesting corollaries and a generalization are also presented.

Reviewer: Eugen Pascu (Montréal)

MSC:

22D10	Unitary representations of locally compact groups
18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
32M05	Complex Lie groups, group actions on complex spaces
32M25	Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions

Keywords:

Kazhdan property; Stein manifold; special linear group