A sixteen-relator presentation of an infinite hyperbolic Kazhdan group

Abstract

We provide an explicit presentation of an infinite hyperbolic Kazhdan group with $4$ generators and $16$ relators of length at most $73$. That group acts properly and cocompactly on a hyperbolic triangle building of type $(3,4,4)$. We also point out a variation of the construction that yields examples of lattices in $\tilde A_2$-buildings admitting non-Desarguesian residues of arbitrary prime power order.