On centralizers of elements of groups acting on trees withinversions

Abstract

A subgroup H of a group G is called malnormal in G if it satisfies the condition that if g ∈ G and h ∈ H, h ≠ 1 such that ghg−1 ∈ H, then g ∈ H. In this paper, 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 an infinite cyclic if g does not transfer an edge of X to its inverse. Otherwise, C(g) is a finite cyclic of order 2.