On centralizers of elements of groups acting on trees with inversions. (Q1811902)

Summary: A subgroup \(H\) of a group \(G\) is called malnormal in \(G\) if it satisfies the condition that if \(g\in G\) and \(h\in H\), \(h\neq 1\), such that \(ghg^{-1}\in H\), then \(g\in H\). We show that if \(G\) is a group acting on a tree \(X\) with inversions such that each edge stabilizer is malnormal in \(G\), then the centralizer \(C(g)\) of each nontrivial element \(g\) of \(G\) is in a vertex stabilizer if \(g\) is in that vertex stabilizer. If \(g\) is not in any vertex stabilizer, then \(C(g)\) is infinite cyclic if \(g\) does not transfer an edge of \(X\) to its inverse. Otherwise, \(C(g)\) is finite cyclic of order 2.