On the existence of nilsolitons on 2-step nilpotent Lie groups

TL;DR

This paper investigates the existence of nilsoliton metrics on 2-step nilpotent Lie groups, showing that many such groups, especially those associated with certain graph structures, do not admit these metrics within specific parameter ranges.

Contribution

It proves the non-existence of nilsoliton metrics on a broad class of 2-step nilpotent Lie groups, extending previous results by Jablonski through graph-based constructions.

Findings

Existence of indecomposable 2-step nilpotent Lie groups without nilsoliton metrics for certain parameters.

Improvement of previous non-existence results by Jablonski.

Identification of parameter ranges where nilsolitons do not exist.

Abstract

A 2-step nilpotent Lie algebra n is said to be of type (p,q)if dim(n)=p+q and dim([n,n])=p. By considering a class of 2-step nilpotent Lie algebras naturally attached to graphs, we prove that there exist indecomposable, 2-step nilpotent Lie groups of type (p,q) which do not admit a nilsoliton metric for every pair (p,q) such that 20 < q and q-2 < p < 1/2q^2-5/2q+10. This improves a result due to Jablonski.