Bulletin of the Belgian Mathematical Society - Simon Stevin

On Kazhdan's Property (T) for the special linear group of holomorphic functions

Björn Ivarsson and Frank Kutzschebauch

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Abstract

We investigate when the group $\mbox{SL}_n(\mathcal{O}(X))$ of holomorphic maps from a Stein space $X$ to $\mbox{SL}_n (\C)$ has Kazhdan's property (T) for $n\ge 3$. This provides a new class of examples of non-locally compact groups having Kazhdan's property (T). In particular we prove that the holomorphic loop group of $\mbox{SL}_n (\C)$ has Kazhdan's property (T) for $n\ge 3$. Our result relies on the method of Shalom to prove Kazhdan's property (T) and the solution to Gromov's Vaserstein problem by the authors.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 21, Number 1 (2014), 185-191.

Dates
First available in Project Euclid: 11 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1394544304

Digital Object Identifier
doi:10.36045/bbms/1394544304

Mathematical Reviews number (MathSciNet)
MR3178540

Zentralblatt MATH identifier
1315.22006

Subjects
Primary: 18F25: Algebraic $K$-theory and L-theory [See also 11Exx, 11R70, 11S70, 12- XX, 13D15, 14Cxx, 16E20, 19-XX, 46L80, 57R65, 57R67] 32M05: Complex Lie groups, automorphism groups acting on complex spaces [See also 22E10]
Secondary: 22D10: Unitary representations of locally compact groups 32M25: Complex vector fields

Keywords
Kazhdan property Stein manifold special linear group

Citation

Ivarsson, Björn; Kutzschebauch, Frank. On Kazhdan's Property (T) for the special linear group of holomorphic functions. Bull. Belg. Math. Soc. Simon Stevin 21 (2014), no. 1, 185--191. doi:10.36045/bbms/1394544304. https://projecteuclid.org/euclid.bbms/1394544304


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