A refinement of the toral rank conjecture for 2-step nilpotent Lie algebras

Abstract:

It is known that the total (co)-homoloy of a 2-step nilpotent Lie algebra $\mathfrak {g}$ is at least $2^{|\mathfrak {z}|}$, where $\mathfrak {z}$ is the center of $\mathfrak {g}$. We improve this result by showing that a better lower bound is $2^t$, where $t={|\mathfrak {z}|+\left [\frac {|v|+1}2\right ]}$ and $v$ is a complement of $\mathfrak {z}$ in $\mathfrak {g}$. Furthermore, we provide evidence that this is the best possible bound of the form $2^t$.