Finite index and finite codimension
Let \(G\) be a finitely generated free group. If \(H\) is a finitely generated subgroup of \(G\), then it is shown that there exists a subgroup \(\text{root}_G(H)\), the root of \(H\) in \(G\), containing \(H\), in which \(H\) has finite index and which contains every subgroup with this property. The proof is based on a theorem of \textit{M. Hall} [Trans. Am. Math. Soc. 67, 421-432 (1949; Zbl 0035.01301)] which asserts the existence of complements \(K\) with the property that the natural map \(H*K\to H\vee K\), the subgroup generated by \(H\) and \(K\), is an isomorphism and the index \((G:H\vee K)\) is finite. Analogous theory of roots of ideals in free algebra \(k\langle X\rangle\) and free group ring \(k(G)\) over a field \(k\) is developed. The author defines the infimum, when \(K\) runs over all Hall complements of \(H\), to be the Hall index \((G:H)_{\text{Hall}}\) of \(H\) in \(G\) and proves that if \(A\) and \(B\) are finitely generated subgroups of \(G\), then \[ \text{rank}(A\cap B)\leq 1+\min\{(G:A)_{\text{Hall}}\cdot(\text{rank}(B)-1),(G:B)_{\text{Hall}}\cdot(\text{rank}(A)-1)\}. \]
scientific article; zbMATH DE number 825328
scientific article