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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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.

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

First available in Project Euclid: 11 March 2014

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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

Kazhdan property Stein manifold special linear group


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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