Existence of cocompact lattices in Lie groups with a bi-invariant metric of index 2

TL;DR

This paper investigates the conditions under which certain simply-connected, indecomposable, solvable Lie groups with bi-invariant metrics of signature (2,n-2) admit cocompact lattices, linking the problem to Salem numbers.

Contribution

It provides a necessary and sufficient criterion for the existence of cocompact lattices in these Lie groups based on their parameters, extending understanding of lattice existence in such geometric structures.

Findings

Lattice existence characterized by parameters of Lie groups

Connection established between lattice existence and Salem numbers

Conditions depend on the dimension of the group's center

Abstract

We study the existence of cocompact lattices in Lie groups with bi-invariant metric of signature $(2,n-2)$. We assume in addition that the Lie groups under consideration are simply-connected, indecomposable and solvable. Then their centre is one- or two-dimensional. In both cases, a parametrisation of the set of such Lie groups is known. We give a necessary and sufficient condition for the existence of a lattice in terms of these parameters. For groups with one-dimensional centre this problem is related to Salem numbers.