Author
R. M. S. Mahmood
Bio: R. M. S. Mahmood is an academic researcher. The author has contributed to research in topics: Cycle graph (algebra) & CA-group. The author has an hindex of 1, co-authored 1 publications receiving 4 citations.
Papers
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TL;DR: In this article, the subgroup theorem for groups acting on======trees with inversions was extended to quasi-HNN groups, and the main technique used is the sub-group theorem.
Abstract: 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.
4 citations
Cited by
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01 Jan 1989
1,062 citations
TL;DR: It is shown 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 .
Abstract: 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 .
1 citations
TL;DR: 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.
28 Sep 2015
TL;DR: In this paper, it was shown that if a group acting on a tree can transfer an edge of a tree into its inverse, then the group is an accessible group on the fiber tree.
Abstract: Throughout this paper the actions of groups on graphs with inversions are allowed. An element g of a group \(G\) is called inverter if there exists a tree \(X\) where \(G\) acts such that \(g\) transfers an edge of \(X\) into its inverse. \(A\) group \(G\) is called accessible if \(G\) is finitely generated and there exists a tree on which \(G\) acts such that each edge group is finite, no vertex is stabilized by $G$, and each vertex group has at most one end. In this paper we show that if \(G\) is a group acting on a tree \(X\) such that if for each vertex \(v\) of \(X\), the vertex group \(G_{v}\) of \(v\) acts on a tree \(X_{v}\), the edge group \(G_{e}\) of each edge e of \(X\) is finite and contains no inverter elements of the vertex group \(G_{t(e)}\) of the terminal \(t(e)\) of $e$, then we obtain a new tree denoted \(\widetilde{X}\) and is called a fiber tree such that \(G\) acts on \(\widetilde{X}\). As an application, we show that if $G$ is a group acting on a tree \(X\) such that the edge group \(G_{e}\) for each edge \(e\) of \(X\) is finite and contains no inverter elements of \(G_{t(e)}\), the vertex \(G_{v}\) group of each vertex \(v\) of \(X\) is accessible, and the quotient graph \(G\diagup X\) for the action of \(G\) on \(X\) is finite, then \(G\) is an accessible group.