Controlled topology invariants of translation actions

We develop invariants Ω n of a translation action of a group on R m analogous to the Bieri–Neumann–Strebel–Renz invariants Σ n . The invariants Σ n were defined to be the set of “directions” e ∈ ∂ ∞ R m such that a suitable universal G-space is ( n − 1 ) -connected over the half-spaces defined by e. We replace half-spaces by topologically more natural neighborhoods of e to obtain the new invariants Ω n . The invariants Σ n and Ω n are related as follows: e ∈ Ω n if and only if every e ′ in an open π 2 -neighborhood of e lies in Σ n .