Extreme events and impact statistics for unipotent actions on the space of lattices

This paper extends a recent extreme value law for horocycle flows on the space of two-dimensional lattices, due to Kirsebom and Mallahi-Karai, to the simplest examples of rank-$k$ unipotent actions on the space of $n$-dimensional lattices. We analyse the problem in terms of the hitting time and impact statistics for the unipotent action with respect to a shrinking surface of section, following the strategy of Pollicott and the first named author in the case of hyperbolic surfaces. If $k=n-1$, the limit law is given by directional statistics of Euclidean lattices, whilst for $k<n-1$ we observe new distributions for which we derive precise tail asymptotics.