Slow entropy of higher rank abelian unipotent actions

Abstract

We study slow entropy invariants for abelian unipotent actions $ U $ on any finite volume homogeneous space $ G/\Gamma $. For every such action we show that the topological slow entropy can be computed directly from the dimension of a special decomposition of $ {{\rm{Lie}}}(G) $ induced by $ {{\rm{Lie}}}(U) $. Moreover, we are able to show that the metric slow entropy of the action coincides with its topological slow entropy. As a corollary, we obtain that the complexity of any abelian horocyclic action is only related to the dimension of $ G $. This generalizes the rank one results from [14] to higher rank abelian actions.