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

TL;DR

This paper explores when groups of holomorphic functions into SL_n have Kazhdan's property (T), introducing new non-locally compact examples and focusing on holomorphic loop groups for n≥3.

Contribution

It establishes Kazhdan's property (T) for groups of holomorphic maps into SL_n, including holomorphic loop groups, using Shalom's method and Gromov's Vaserstein problem solution.

Findings

Holomorphic loop group of SL_n has Kazhdan's property (T) for n≥3

Provides new non-locally compact examples of groups with property (T)

Utilizes novel combination of existing methods and solutions

Abstract

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.