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

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

2012-02-29

arxiv.org/abs/1202.6531v1

Mathematics

Complex Variables

5 pages

Scientific paper

We investigate when the group $SL_n(\mathcal{O}(X))$ of holomorphic maps from a Stein space $X$ to $SL_n (\C)$ 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 $SL_n (\C)$ 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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