The characteristic group of Lie LCP manifolds

TL;DR

This paper investigates the structure of the characteristic group in Lie locally conformally product (LCP) manifolds, establishing that it is contained in the radical of the Lie group and thus simply connected, advancing understanding of LCP geometry.

Contribution

It proves that the reduced characteristic group of Lie LCP manifolds is contained in the radical of the Lie group, showing it is simply connected, which was previously unknown.

Findings

The characteristic group is contained in the radical of the Lie group.

The characteristic group is simply connected.

Provides new insights into the structure of Lie LCP manifolds.

Abstract

The (reduced) characteristic group of a locally conformally product manifold is obtained by restricting the action of its fundamental group to the non-flat factor of the universal cover, and taking the connected component of the identity in the closure of this restriction. It was shown by Kourganoff that this group is abelian, but it is currently unknown whether it is simply connected, or might have compact (toric) factors. This question is crucial for a better understanding of LCP structure, as shown recently by B. Flamencourt. In this paper we consider Lie LCP structures (which are defined on quotients of simply connected Lie groups by lattices) and show that the reduced characteristic group of any Lie LCP manifold $\Gamma\backslash G$ is contained in the radical of $G$, so in particular is simply connected.