We extend the structure theorem for the subgroups of the class of
HNN groups to a new class of groups called quasi-HNN groups. The
main technique used is the subgroup theorem for groups acting on
trees with inversions.

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Subgroups of quasi-HNN groups

In this paper we show that if G is a group acting on a graph X with inversions such that G has a presentation induced by a fundamental domain for the action of G on X, then X is a tree.

On the Converse of the Theory of Groups Acting on Trees with Inversions

A graph is called a quasi graph if the possibility of an edge of the graph being equal to its inverse is not excluded. Quasi HNN groups are a new generalizations of HNN groups. In this paper we introduce the concepts of a quasi graph of groups and its fundamental group, and show that the fundamental group of a quasi graph of groups is a quasi HNN group. The embedding theorem for the fundamental group of a quasi graph of groups is formulated and proved. Furthermore, we find the structures of groups induced by the vertices of a quasi graph of groups.

/pdf/fundamental-groups-of-quasi-graphs-of-groups-2gz3dzgl0v.pdf

Fundamental Groups of Quasi Graphs of Groups

Quasi-HNN groups can be characterized as a generalization of HNN
groups. In this paper, we show that if G ∗ is a quasi-HNN
group of base G , then either any two conjugates of G are
identical or their intersection is contained in a conjugate of
an associated subgroup of G .

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A remark on the intersection of the conjugates of the base of quasi-hnn groups

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.

R.M.S. Mahmood

Papers

On quasi HNN groups

Structures of Fibers of Groups Actions on Graphs