On invertor elements and finitely generated subgroups of groups acting on trees with inversions (Q1578340)

Summary: An element of a group acting on a graph is called invertor if it transfers an edge of the graph to its inverse. In this paper, we show that if \(G\) is a group acting on a tree \(X\) with inversions such that \(G\) does not fix any element of \(X\), then an element \(g\) of \(G\) is invertor if and only if \(g\) is not in any vertex stabilizer of \(G\) and \(g^2\) is in an edge stabilizer of \(G\). Moreover, if \(H\) is a finitely generated subgroup of \(G\), then \(H\) contains an invertor element or some conjugate of \(H\) contains a cyclically reduced element of length at least one on which \(H\) is not in any vertex stabilizer of \(G\), or \(H\) is in a vertex stabilizer of \(G\).