Augmented group systems and shifts of finite type
Given an epimorphism \(\chi\colon G\to\mathbb{Z}\) of the finitely generated group \(G\) onto the integers, the authors investigate \(K_\chi=\text{kernel} \chi\). The situation arises for \(G\) a knot group and \(K_\chi\) its commutator subgroup; it is well known that \(K_\chi\) might or might not be finitely generated. The main result is: \(K_\chi\) contains finitely many subgroups \(H\) with index \([K_\chi:H]\leq r\) for any \(r<\infty\), if \(\chi\) represents an element of \((\Hom(G,\mathbb{R})-\{0\})/\mathbb{R}_+\) which belongs to the Bieri-Neumann-Strebel subset \(\Sigma\). The argument uses the sets \(\phi_r\) of representations \(\rho\colon\text{Kernel} \chi\to S_r\) (= symmetric group) which can be related to the subgroups \(H\) by covering theory, and shifts \[ \sigma_x\colon\phi_r\to\phi_r,\quad(\sigma_x\rho)a=\rho(x^{-1}ax)\text{ for }x\in\chi^{-1}(1). \] From the shifts an entropy invariant can be derived with application to knot theory. A corollary to the main theorem states that \(K_\chi\) is Hopfian if \(G\) is residually finite and \(\chi\) in \(\Sigma\). As a counterexample the authors show that the commutator subgroup of the group of the knot \(5_2\) has uncountably many subgroups of index \(r>3\).
scientific article; zbMATH DE number 952565
scientific article