Quaternionic hyperbolic lattices of minimal covolume

TL;DR

This paper identifies the smallest covolume lattices in the Lie group Sp(n,1) for all n>1, providing explicit descriptions using Hurwitz and icosian rings, advancing understanding of quaternionic hyperbolic lattices.

Contribution

It explicitly determines minimal covolume lattices in Sp(n,1) and describes them using specific algebraic rings, a novel classification for both uniform and nonuniform cases.

Findings

Explicit descriptions of minimal covolume lattices for all n>1.

Use of Hurwitz integers for nonuniform lattices with even n.

Use of icosian ring for uniform lattices for all n>1.

Abstract

For any n>1 we determine the uniform and nonuniform lattices of the smallest covolume in the Lie group Sp(n,1). We explicitly describe them in terms of the ring of Hurwitz integers in the nonuniform case with n even, respectively, of the icosian ring in the uniform case for all n>1.