Lattices in contact Lie groups and 5-dimensional contact solvmanifolds

TL;DR

This paper classifies 5-dimensional contact Lie groups with uniform lattices, showing there are exactly seven such solvable groups and exploring related geometric structures and boundary issues.

Contribution

It provides a complete classification of 5D contact Lie groups with uniform lattices, extending Alexander's criteria and analyzing their geometric properties.

Findings

Seven 5D contact Lie groups admit uniform lattices

All such groups are solvable

The special affine group has no uniform lattice

Abstract

This paper investigates the geometry of compact contact manifolds that are uniformized by contact Lie groups, i.e., compact manifolds that are the quotient of some Lie group G with a left invariant contact structure and a uniform lattice subgroup. We re-examine Alexander's criteria for existence of lattices on solvable Lie groups and apply them, along with some other well known tools, and use these results to prove that, in dimension 5, there are exactly seven connected and simply connected contact Lie groups with uniform lattices, all of which are solvable. Issues of symplectic boundaries are explored, as well. It is also shown that the special affine group has no uniform lattice.