On the topological invariants and for extensions of (Lie groups over a p-adic field)-by-abelian groups

For G a locally compact group and i=1,2 we define topological versions of the geometric homotopical invariants Σ1 and Σ2 of discrete groups. We calculate and for G=exp(η) Q, η a nilpotent Lie algebra over a local p-adic field K and Q an abstract free abelian group of finite rank that acts on expη via topological automorphisms. An important part of the structure of η is that it splits as a direct sum of one-dimensional (over K) K[Q]-modules. We conjecture the structure of the Bieri–Strebel–Renz invariant Σ2(H) for a discrete nilpotent-by-abelian S-arithmetic group H. The invariant Σ2(H) characterizes the finitely presented subgroups of H that contain the commutator.