Factorization of Holomorphic Matrices and Kazhdan's property (T)

TL;DR

This paper explores algebraic properties of holomorphic symplectic matrices on Stein spaces, establishing factorization results and demonstrating that certain groups possess Kazhdan's property (T), with explicit bounds provided.

Contribution

It introduces explicit bounds for holomorphic and exponential factorizations and proves that the elementary symplectic group has Kazhdan's property (T).

Findings

Explicit bounds for holomorphic factorization on low-dimensional Stein spaces

Bounds for exponential factorization derived from holomorphic bounds

Elementary symplectic group admits Kazhdan's property (T)

Abstract

In this article we deduce some algebraic properties for the group $\mathrm{Sp}_{2n} (\mathcal{O}(X))$ of holomorphic symplectic matrices on a Stein space $X$: holomorphic factorization, exponential factorization, and Kazhdan's property (T). In holomorphic factorization we combine a recent result of the third author and K-theory tools to give explicit bounds for the case when $X$ is one-dimensional or two-dimensional. Next we use them to find bounds for exponential factorization. As a further application, we show that the elementary symplectic group $\mathrm{Ep}_{2n}(\mathcal{O}(X))$ admits Kazhdan's property (T).