## Bulletin of the Belgian Mathematical Society - Simon Stevin

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

#### 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

https://projecteuclid.org/euclid.bbms/1394544304

Digital Object Identifier
doi:10.36045/bbms/1394544304

Mathematical Reviews number (MathSciNet)
MR3178540

Zentralblatt MATH identifier
1315.22006

#### 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