Subgroups of quasi-HNN groups

On centralizers of elements of groups acting on trees with inversions

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.

On fibers and accessibility of groups acting on trees with inversions

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.

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

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 .

Combinatorial Group Theory

Abstract: Chapter I. Free Groups and Their Subgroups 1. Introduction 2. Nielsen's Method 3. Subgroups of Free Groups 4. Automorphisms of Free Groups 5. Stabilizers in Aut(F) 6. Equations over Groups 7. Quadratic sets of Word 8. Equations in Free Groups 9. Abstract Lenght Functions 10. Representations of Free Groups 11. Free Pruducts with Amalgamation Chapter II. Generators and Relations 1. Introduction 2. Finite Presentations 3. Fox Calculus, Relation Matrices, Connections with Cohomology 4. The Reidemeister-Schreier Method 5. Groups with a Single Defining Relator 6. Magnus' Treatment of One-Relator Groups Chapter III. Geometric Methods 1. Introduction 2. Complexes 3. Covering Maps 4. Cayley Complexes 5. Planar Caley Complexes 6. F-Groups Continued 7. Fuchsian Complexes 8. Planar Groups with Reflections 9. Singular Subcomplexes 10. Sherical Diagrams 11. Aspherical Groups 12. Coset Diagrams and Permutation Representations 13. Behr Graphs Chpter IV. Free Products and HNN Extensions 1. Free Products 2. Higman-Neumann-Neumann Extensions and Free Products with Amalgamation 3. Some Embedding Theorems 4. Some Decision Problems 5. One-Relator Groups 6. Bipolar Structures 7. The Higman Embedding Theorem 8. Algebraically Closed Groups Chapter V. Small Cancellation Theory 1. Diagrams 2. The Small Cancellation Hypotheses 3. The Basic Formulas 4. Dehn's Algorithm and Greendlinger's Lemma 5. The Conjugacy Problem 6. The Word Problem 7. The Cunjugacy Problme 8. Applications to Knot Groups 9. The Theory over Free Products 10. Small Cancellation Products 11. Small Cancellation Theory over free Products with Amalgamation and HNN Extensions Bibliography Index of Names Subject Index

Subgroups of HNN groups

Abstract: The purpose of this paper is to give a more precise form of Theorem 1 of [2], which gives a structure theorem for subgroups of HNN groups; we prove the following.Let H be a subgroup of the HNN group . Then H is an HNN group whose base is a tree product of groups H ∪ wAw-1 where w runs over a set of double coset representatives of (H,A); the amalgamated and associated subgroups are all of the form H ∊ vUiv-l for some v. We can be more precise about which subgroups occur and about the tree product. We will also obtain stronger forms of other results in [1] and [2].