Directional expansion in ergodic actions of countable groups

Abstract

We study directional expansion for probability-measure-preserving actions of countable groups through a representation-theoretic group property, the cyclic escape property. An infinite countable group has the cyclic escape property if every totally ergodic unitary representation has arbitrarily small fixed-vector projections along infinite cyclic subgroups. This property implies directional expansivity for all totally ergodic actions. We prove that all infinite finitely generated nilpotent groups have the cyclic escape property, and conjecture the same for all infinite finitely generated polycyclic groups. We also prove the cyclic escape property for higher-rank simple lattices whose finite-dimensional unitary representations all have finite image; in particular, for $SL_n(\mathbb Z)$, $PSL_n(\mathbb Z)$, and $PGL_n(\mathbb Z)$, $n\geq 3$. By contrast, free groups of rank at least two do not have the cyclic escape property. The proofs exhibit two independent mechanisms: central spectral structure in nilpotent groups and stationary character rigidity in higher-rank lattices.