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 R. M. S. Mahmood and M. I. Khanfar. 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 is a group acting on a tree with inversions
such that does not fix any element of , then an element of is invertor if and only if is not in any vertex
stabilizer of and is in an edge stabilizer of . Moreover, if is a finitely generated subgroup of , then contains an invertor element or some conjugate of contains a
cyclically reduced element of length at least one on which is
not in any vertex stabilizer of , or is in a vertex
stabilizer of .