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

We investigate when the group $S{L}_n(\mathcal{O}(X))$ of holomorphic maps from a Stein space $X$ to $S{L}_n(\text{C})$ 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 $SL_n (\C)$hasKazhda{n}^{\mathrm{\prime }}sproperty(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.