On Low-Dimensional Solvmanifolds

TL;DR

This paper investigates the existence of lattices in low-dimensional solvable Lie groups and examines their associated solvmanifolds for properties like formality, symplecticity, and Kählerness, highlighting the complexities compared to nilmanifolds.

Contribution

It provides a systematic analysis of lattices in solvable Lie groups up to dimension six and explores their geometric and topological properties.

Findings

Lattices exist in certain low-dimensional solvable Lie groups.

Some solvmanifolds are formal and symplectic.

The criteria for lattices in solvable groups are more complex than for nilpotent groups.

Abstract

A nilmanifold resp. solvmanifold is a compact homogeneous space of a connected and simply-connected nilpotent resp. solvable Lie group by a lattice, i.e. a discrete co-compact subgroup. There is an easy criterion for nilpotent Lie groups which enables one to decide whether there is a lattice or not. Moreover, it is easy to decide whether a nilmanifold is formal, Kaehlerian or (Hard) Lefschetz. The study of solvmanifolds meets with noticeably greater obstacles than the study of nilmanifolds. Even the construction of solvmanifolds is considerably more difficult than is the case for nilmanifolds. The reason is that there is no simple criterion for the existence of a lattice in a connected and simply-connected solvable Lie group. We consider the question of existence of lattices in solvable Lie groups up to dimension six and examine whether the corresponding solvmanifolds are formal, symplectic, Kaehler or (Hard) Lefschetz.