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International Journal of Mathematics and Mathematical Sciences
Volume 23 (2000), Issue 9, Pages 585-595

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

1Ajman University of Science and Technology, Abu Dhabi, United Arab Emirates
2Department of Mathematics, Yarmouk University, Irbid, Jordan

Received 9 February 1999

Copyright © 2000 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 g2 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.