SL2 representations and relative property (T)

Abstract

We investigate relative property (T) of Kazhdan-Margulis for (SL2(R)⋉Rn,Rn), mainly where R is a discrete finitely generated commutative ring. The action of SL2(R) on Rn is defined through admissible lattices of the irreducible representations of sl2(C). Both positive and negative results are obtained. Applications of these results will be given, including the establishment of property (T) for certain Kac-Moody type groups over commutative rings.