Infra-solvmanifolds of $\Sol_1^4$-geometry

TL;DR

This paper classifies all compact manifolds with $Sol_1^4$-geometry, detailing their holonomy groups and including the classification of 3-dimensional infra-$Sol$ manifolds, thereby expanding understanding of these geometric structures.

Contribution

It provides a complete classification of infra-$Sol_1^4$-manifolds, including explicit examples with all possible holonomy groups, and extends the classification to 3-dimensional infra-$Sol$ manifolds.

Findings

All holonomy groups occur in infra-$Sol_1^4$-manifolds.

Explicit example of an infra-$Sol_1^4$-manifold with $D_4$ holonomy.

Includes classification of 3-dimensional infra-$Sol$ manifolds.

Abstract

The purpose of this paper is to classify all compact manifolds modeled on the 4-dimensional solvable Lie group $Sol_1^4$. The maximal compact subgroup of $Isom(Sol_1^4)$ is $D_4=\mathbb Z_4\rtimes\mathbb Z_2$. We shall exhibit an infra-solvmanifold with $Sol_1^4$-geometry whose holonomy is $D_4$. This implies that all possible holonomy groups do occur; $\{1\}$, $\mathbb Z_2$ (5 families), $\mathbb Z_4$, $\mathbb Z_2\times\mathbb Z_2$ (5 families),and $\mathbb Z_4\rtimes\mathbb Z_2$ (2 families). This includes the classification of 3-dimensional infra-$Sol$ manifolds.