Nonarchimedean superrigidity of solvable \(S\)-arithmetic groups. (Q404242)

Let \(\Gamma\) be an \(S\)-arithmetic subgroup of a solvable algebraic group \(\mathbf G\) over an algebraic number field \(F\), such that the finite set \(S\) contains at least one place that is non-archimedean. The authors construct a certain group \(H\) such that if \(L\) is any local field and \(\alpha\colon\Gamma\to\mathrm{GL}_n(L)\) is any homomorphism, then \(\alpha\) virtually extends (modulo a bounded error) to a continuous homomorphism defined on some finite-index subgroup of \(H\). In the special case where \(F=\mathbb Q\), \(\mathrm{rank}_{\mathbb R}\mathbf G=0\), and \(\Gamma\) is Zariski-dense in \(\mathbf G\), we may set \(H=\mathbf G_S\). The authors also point out a generalization that does not require \(\mathbf G\) to be solvable.