A sixteen-relator presentation of an infinite hyperbolic Kazhdan group

TL;DR

This paper constructs an explicit example of an infinite hyperbolic Kazhdan group with specific generators and relators, demonstrating its action on a hyperbolic triangle building and exploring related lattice structures in $ ilde A_2$-buildings.

Contribution

It provides a new explicit presentation of an infinite hyperbolic Kazhdan group and explores its geometric and lattice-theoretic properties.

Findings

Constructed a 16-relator hyperbolic Kazhdan group

Demonstrated proper and cocompact action on a hyperbolic triangle building

Presented variations yielding lattices in $ ilde A_2$-buildings with non-Desarguesian residues

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.