torsion-free Abelian groups of finite rank
Torsion-free Abelian groups defined by an integral matrix.
Let \(\varphi\) be an injective endomorphism of a free Abelian group \(F\) of finite rank \(k\). Then the group \(G(\varphi,F)=\bigcup_{n\geq 0}\varphi^{-n}(F)\) is a torsion-free group of rank \(k\).NEWLINENEWLINE The author investigates the properties of such groups in terms of the eigenvalues of \(\varphi\) in the algebraic closure of \(\mathbb Q\). His paper contains many novel results on the structure of tffr groups, of which the following is the most remarkable:NEWLINENEWLINE Let \(G\) be a torsion-free group of rank \(k\). Then \(G\cong G(\varphi,F)\) for a free group \(F\) of rank \(k\) and an injective diagonalizable endomorphism \(\varphi\) of \(F\) if and only if \(G\) is almost completely decomposable and all the critical types of \(G\) are of ring type and divisible by only finitely many primes.
scientific article; zbMATH DE number 6081654