On centralizers of elements of groups acting on trees with inversions

TLDR: In this paper, the authors 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.

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 g h g − 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.