Solvable Baumslag-Solitar Lattices

The solvable Baumslag Solitar groups BS(1,n) each admit a canonical model space, Xₙ. We give a complete classification of lattices in Gₙ = Isom⁺(Xₙ) and find that such lattices fail to be strongly rigidx2014there are automorphisms of lattices Γ ⊂ Gₙ which do not extend to Gₙ{x2014}$but do satisfy a weaker form of rigidity: for all isomorphic lattices $Γ_1,Γ_2⊂ G_n$, there is an automorphism $ρ ∈ Aut(G_n)$ so that $ρ(Γ_1) = Γ_2$.