Solvable Baumslag-Solitar Lattices

The solvable Baumslag Solitar groups $\text{BS}(1,n)$ each admit a canonical model space, $X_n$. We give a complete classification of lattices in $G_n={\text{Isom}}^{+}(X_n)$ and find that such lattices fail to be strongly rigid$\text{unicode}x2014$there are automorphisms of lattices $\mathrm{\Gamma }\subset G_n$ which do not extend to $G_n$$\text{unicode}x2014$but do satisfy a weaker form of rigidity: for all isomorphic lattices ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2\subset G_n$, there is an automorphism $\rho \in \text{Aut}(G_n)$ so that $\rho ({\mathrm{\Gamma }}_1)={\mathrm{\Gamma }}_2$.