As the title indicates the authors announce a rigidity result on solvable arithmetic groups. Following the authors, define a solvable linear algebraic group \(G\) over \(Q\) to be reduced if the centralizer in \(G\) of the unipotent radical \(R_u(G)\) of \(G\) is contained in \(R_u(G)\) itself.
The theorem proved in this paper is Theorem 1. If \(\Gamma_1\) and \(\Gamma_2\) are arithmetic Zariski dense subgroups of reduced solvable algebraic groups \(G_1\) and \(G_2\) and \(\varphi\colon\Gamma_1\to\Gamma_2\) a rational isomorphism, then there exist a rational homomorphism \(\Phi\colon G_1\to G_2\) defined over \(Q\) and a set theoretic map \(u\colon\Gamma_1\to Z(G_2^0)(Q)\) (= rational points of the centre of the connected component of identity in \(G_2\)) such that \(\Phi(r)=\varphi(\gamma)u(r)\) i.e., upto centre, \(\Phi\) is an extension of the group homomorphism \(\varphi\).
The authors also state a uniform version of this rigidity result. Let \(\operatorname{Aut}(G)_Q\) denote the group of \(Q\)-rational automorphisms of \(G\), \(\operatorname{Aut}(\Gamma)\) the group of abstract automorphisms of \(\Gamma\), \(Z(G)/(Q)\) the rational points of the centre of \(G\).
Theorem 2. Let \(G\) be a connected reduced solvable algebraic group over \(Q\) and \(\Gamma\subset G(Q)\) an arithmetic subgroup. There is a homomorphism \[ D\colon\operatorname{Aut}(\Gamma)\to\operatorname{Aut}_Q(G)\quad (\varphi\mapsto D_\varphi) \] and a map \(u\colon\operatorname{Aut}(\Gamma)\to\operatorname{Hom}(\Gamma,Z(G)(Q))\) (\(\varphi\mapsto u_\varphi\)) such that for all \(\varphi\in\operatorname{Aut}(\Gamma)\), and \(\gamma\in\Gamma\), we have \(D_\varphi(\gamma)=u_\varphi(\gamma)\varphi(\gamma)\).
Observe that Theorems 1 and 2 are direct extensions (to the solvable case) of results of Mostow, Prasad and Margulis on rigidity of lattices in semisimple centreless groups (in the semisimple case, the maps \(u\) and \(u_\varphi\) are identically one). The authors also give a number of interesting applications of Theorem 2.