Logical aspects of the theory of rigid solvable groups
N. Romanovskiy
A group \(G\) is said to be \(m\)-rigid if it has a normal series
\[G=G_1 > G_2 > \ldots > G_m > G_{m+1}=1\] with abelian factors each of which,
\(G_i/G_{i+1}\), viewed as a \(\mathbb{Z} [G/G_i]\)-module, has no torsion.
Free solvable groups are rigid. A rigid group \(G\) is said to be divisible if any factor \(G_i/G_{i+1}\) is a divisible module over the ring \(\mathbb{Z} [G/G_i]\) or, in other words, it is a vector space over the skew field of fractions of this ring.
We say, for \(m\)-rigid groups, that \(H\) is embedded into \(G\) with preserving linear independence, if any system of elements of \(H_i/H_{i+1}\) linearly independent over \(\mathbb{Z} [H/H_i]\) is linearly independent over \(\mathbb{Z} [G/G_i]\).
Theorem 1.Arbitrary \(m\)-rigid group can be embedded with preserving linear independence into some divisible \(m\)-rigid group.
Malcev proved that a free solvable group of length \(\geq 2\) has undecidable elementary theory. The universal theory of a free metabelian group is decidable (Chapuis).
Theorem 2.The universal theory of a free solvable group of length \(\geq 4\) is undecidable.
For the class \(\Sigma_m\) of rigid groups of length \(\leq m\) we define algebraically closed objects: \(G\) is said to be algebraically closed
if, for any embedding \(G \hookrightarrow H\) in this class with preserving linear independence, any system of equations over \(x_1, \ldots ,x_n\) with coefficients in \(G\) has a solution in \(G^n\) if and only if it has a solution in \(H^n.\) \(G\) is said to be existentially closed if, for any such embedding, any \(\exists\)-formula is true on \(G\) if and only if it is true on \(H.\)
Theorem 3.The algebraically closed groups in \(\Sigma_m\) are precisely the divisible \(m\)-rigid groups. They are also existentially closed objects in \(\Sigma_m.\)
We study elementary theories of divisible \(m\)-rigid groups, and construct a system of axioms in group theory signature which defines exactly all divisible \(m\)-rigid groups. Denote by \(T\) the corresponding theory.
Fix some countable divisible \(m\)-rigid group \(M.\) We note that this group is constructible. Extend the signature of group theory by constants from \(M.\) We add some recursive system of axioms, which means that \(M\) is embedded into a given rigid group with preserving linear independence. Denote the corresponding theory by \(T_M\).
Theorem 4.The theories \(T\) and \(T_M\) are complete and therefore decidable.
Theorems 3 and 4 were proved jointly with Alexei Myasnikov.
