Diophantine problems in solvable groups

Abstract

We study the Diophantine problem (decidability of finite systems of equations) in different classes of finitely generated solvable groups (nilpotent, polycyclic, metabelian, free solvable, etc), which satisfy some natural "non-commutativity" conditions. For each group $G$ in one of these classes, we prove that there exists a ring of algebraic integers $O$ that is interpretable in $G$ by finite systems of equations (e-interpretable), and hence that the Diophantine problem in $O$ is polynomial time reducible to the Diophantine problem in $G$. One of the major open conjectures in number theory states that the Diophantine problem in any such $O$ is undecidable. If true this would imply that the Diophantine problem in any such $G$ is also undecidable. Furthermore, we show that for many particular groups $G$ as above, the ring $O$ is isomorphic to the ring of integers $\mathbb{Z}$, so the Diophantine problem in $G$ is, indeed, undecidable. This holds, in particular, for free nilpotent or free solvable non-abelian groups, as well as for non-abelian generalized Heisenberg groups and uni-triangular groups $UT(n,\mathbb{Z}), n \geq 3$. Then we apply these results to non-solvable groups that contain non-virtually abelian maximal finitely generated nilpotent subgroups. For instance, we show that the Diophantine problem is undecidable in the groups $GL(3,\mathbb{Z}), SL(3,\mathbb{Z}), T(3,\mathbb{Z})$.