Property (T) for Kac-Moody groups over rings

TL;DR

This paper proves that certain Kac-Moody groups over rings have property (T) under specific algebraic conditions on the ring, extending understanding of rigidity properties in infinite-dimensional groups.

Contribution

It establishes new criteria for property (T) in Kac-Moody groups over rings based on ring ideal and invertibility conditions, advancing the theory of these groups.

Findings

Kac-Moody groups G_A(R) have property (T) under specified ring conditions.

Existence of an integer n(A) ensuring property (T) for rings with no small-index ideals.

Conditions relate ring ideal structure and invertibility to rigidity properties of the group.

Abstract

Let R be a finitely generated commutative ring with 1, let A be an indecomposable 2-spherical generalized Cartan matrix of size at least 2 and M=M(A) the largest absolute value of a non-diagonal entry of A. We prove that there exists an integer n=n(A) such that the Kac-Moody group G_A(R) has property (T) whenever R has no proper ideals of index less than n and all positive integers less than or equal to M are invertible in R.