Property (T) for Kac–Moody groups over rings

Abstract

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 GA(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.

Link conditions for cubulation

“…The full automporphism groups of Kac-Moody buildings of 2-spherical type of large thickness have Property (T ) [DJ02,ER18]: neither [DJ02] nor [ER18] record whether Property (T) fails at small thicknesses. Some of the groups considered in Corollary B are cocompact lattices in a 2-spherical Kac-Moody building with small thickness [CKV12].…”

Hyperbolic generalized triangle groups, property (T) and finite simple quotients

“…Remark For $$p$$ odd and $$q$$ any power of $$p$$, the estimate $$$$\begin{equation*} \varepsilon _{\mathcal U_4(q)}(U_1, U_4) \leqslant \sqrt {\frac{ 1 +\sqrt p }{p} } \end{equation*}$$$$can be extracted from the work of Ershov–Rall [31]. …”

“…The following procedure allows one to verify with Magma that 𝐿 is isomorphic 𝑋 ≅ PSL 2 (109). First, one computes that the assignments (31) where the generators are represented by matrices with the same letter. We conclude that the coset graph has girth 14 and that 5𝜀 𝑋 (𝐴, 𝐵) < √ 15 ≈ 3.8729833462.…”

“…Clearly, the method used above yields more examples of hyperbolic 5-fold generalized triangle groups with property (T) than those recorded in Theorem 1.2. We may indeed use PSL 2 (41) instead of PSL 2 (31), and PSL 2 (131) instead of PSL 2 (109), as a consequence of Proposition 2.17. Moreover, for each triple of vertex groups, we can take advantage of the freedom we have in defining the homomorphisms identifying an edge group to a generator of the vertex group containing it.…”