The characteristic group of Lie LCP manifolds
Viviana del Barco, Andrei Moroianu
TL;DR
The paper addresses the structure of the reduced characteristic group $\bar{{\rm P}}_0$ for locally conformally product (LCP) manifolds in the Lie setting. By modeling universal covers as semi-direct products $G=U\rtimes H$ with conformal actions, and employing Levi decompositions and lattice arguments, it shows that $\bar{{\rm P}}_0$ must lie in the radical $R$ of $G$, hence is simply connected. In the solvable (solvmanifold) case, this yields that the characteristic group is simply connected, and in general it shows that $\bar{{\rm P}}_0$ is contained in the commutator subgroup, aligning with observed examples. These results advance Flamencourt’s question within Lie LCP manifolds and provide a pathway toward a classification via the reduced characteristic group.
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 $Γ\backslash G$ is contained in the radical of $G$, so in particular is simply connected.