Weak potency of fundamental groups of graphs of groups.
A group \(G\) is called weakly potent if for any element \(x\) of infinite order in \(G\), we can find a positive integer \(r\) with the property that for every positive integer \(n\), there exists a normal subgroup \(M_n\) of finite index in \(G\) such that \(xM_n\) has order exactly \(rn\). \textit{B. Evans} [Pac. J. Math. 55, 371-379 (1974; Zbl 0327.20013)] and \textit{C. Y. Tang} [Can. Math. Bull. 38, No. 1, 120-127 (1995; Zbl 0822.20027)] proved that polycyclic-by-finite and free-by-finite groups are weakly potent. A group \(G\) is called \(H\)-separable for the subgroup \(H\) if for each \(x\in G\setminus H\), there exists \(N\triangleleft_fG\) such that \(x\not\in HN\). In this paper the authors prove: Theorem: (Th. 3.1 in the paper) Let \(G=\langle t,A\mid t^{-1}ht=k\rangle\) where \(A\) is a weakly potent group and \(\langle h\rangle,\langle k\rangle\) are infinite cyclic subgroups of \(A\) such that \(\langle h\rangle\cap\langle k\rangle\neq 1\). Suppose that \(A\) is \(\langle h\rangle\)-separable and \(\langle k\rangle\)-separable. Then \(G\) is weakly potent if and only if \(h^m=k^{\pm m}\) for some \(m>0\). Theorem: (Th. 4.1 in the paper) Let \(G=G_1*_HG_2\) where \(G_1,G_2\) are weakly potent and \(H=\langle h\rangle\) is infinite cyclic. Suppose that \(G_1,G_2\) are \(\langle h\rangle\)-separable. Then \(G\) is weakly potent. Extending these theorems to the case of HNN extensions with many free generators \(t_i\) and in the case of tree products with many vertices and combining them, the authors give a characterization for the fundamental group of graph of weakly potent groups with infinite cyclic edge groups to be weakly potent. -- As a corollary they prove that the one-relator groups with non-trivial center are weakly potent.
scientific article; zbMATH DE number 5712981