THE COMBINATION THEOREM AND QUASICONVEXITY

2001 ◽  
Vol 11 (02) ◽  
pp. 185-216 ◽  
Author(s):  
ILYA KAPOVICH
Keyword(s):  
Fundamental Group ◽  
Isometric Embedding ◽  
Normal Forms ◽  
Vertex Group ◽  
Graph Of Groups

We show that if G is a fundamental group of a finite k-acylindrical graph of groups where every vertex group is word-hyperbolic and where every edge-monomorphism is a quasi-isometric embedding, then all the vertex groups are quasiconvex in G (the group G is word-hyperbolic by the Combination Theorem of M. Bestvina and M. Feighn). This allows one, in particular, to approximate the word metric on G by normal forms for this graph of groups.

scholarly journals Isometric embeddings in bounded cohomology

2014 ◽  
Vol 06 (01) ◽  
pp. 1-25 ◽  
Author(s):  
M. Bucher ◽  
M. Burger ◽  
R. Frigerio ◽  
A. Iozzi ◽  
C. Pagliantini ◽  
...  
Keyword(s):  
Fundamental Group ◽  
Isometric Embedding ◽  
Equivalence Theorem ◽  
Connected Component ◽  
Bounded Cohomology ◽  
Graph Of Groups ◽  
Simplicial Volume ◽  
Direct Sum ◽  
Isometric Embeddings ◽  
Relative Bounded Cohomology

This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups, we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X, Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X, Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov's Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along π1-injective boundary components with amenable fundamental group.

FIXED POINTS OF SYMMETRIC ENDOMORPHISMS OF GROUPS

2002 ◽  
Vol 12 (05) ◽  
pp. 737-745 ◽  
Author(s):  
MIHALIS SYKIOTIS
Keyword(s):  
Fundamental Group ◽  
Fixed Points ◽  
Structure Theorem ◽  
Graph Of Groups

Let G be the fundamental group of a graph of groups with finite edge groups and f an endomorphism of G. We prove a structure theorem for the subgroup Fix(f), which consists of the elements of G fixed by f, in the case where the endomorphism f of G maps vertex groups into conjugates of themselves.

scholarly journals Parameterizations of 1-Bridge Torus Knots

2003 ◽  
Vol 12 (04) ◽  
pp. 463-491 ◽  
Author(s):  
Doo Ho Choi ◽  
Ki Hyoung Ko
Keyword(s):  
Normal Form ◽  
Fundamental Group ◽  
Explicit Formula ◽  
Normal Forms ◽  
Double Cover ◽  
Torus Knots ◽  
Number Of Components ◽  
Torus Knot ◽  
Branched Cover ◽  
The Family

A 1-bridge torus knot in a 3-manifold of genus ≤ 1 is a knot drawn on a Heegaard torus with one bridge. We give two types of normal forms to parameterize the family of 1-bridge torus knots that are similar to the Schubert's normal form and the Conway's normal form for 2-bridge knots. For a given Schubert's normal form we give algorithms to determine the number of components and to compute the fundamental group of the complement when the normal form determines a knot. We also give a description of the double branched cover of an ambient 3-manifold branched along a 1-bridge torus knot by using its Conway's normal form and obtain an explicit formula for the first homology of the double cover.

scholarly journals Abstract commensurators of surface groups

2019 ◽  
pp. 1-16
Author(s):  
Khalid Bou-Rabee ◽  
Daniel Studenmund
Keyword(s):  
Fundamental Group ◽  
Finite Type ◽  
Computational Methods ◽  
Normal Forms ◽  
Computer Assisted ◽  
Surface Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Hnn Extensions ◽  
Concrete Objects

Let [Formula: see text] be the fundamental group of a surface of finite type and [Formula: see text] be its abstract commensurator. Then [Formula: see text] contains the solvable Baumslag–Solitar groups [Formula: see text] for any [Formula: see text]. Moreover, the Baumslag–Solitar group [Formula: see text] has an image in [Formula: see text] that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of [Formula: see text] are concrete objects amenable to computational methods. For example, we give a proof that [Formula: see text] is not residually finite without the use of normal forms of HNN extensions.

scholarly journals A combination theorem for affine tree-free groups

2016 ◽  
Vol 26 (07) ◽  
pp. 1283-1321
Author(s):  
Shane O. Rourke
Keyword(s):  
Abelian Group ◽  
Fundamental Group ◽  
Large Class ◽  
Recent Work ◽  
Isometric Action ◽  
Graph Of Groups ◽  
Finitely Generated Group ◽  
Affine Action ◽  
Relatively Hyperbolic ◽  
Solvable Word

Let [Formula: see text] be an ordered abelian group. We show how a group admitting a free affine action without inversions on a [Formula: see text]-tree admits a natural graph of groups decomposition, where vertex groups inherit actions on [Formula: see text]-trees. We introduce a stronger condition (essential freeness) on an affine action and apply recent work of various authors to deduce that a finitely generated group admitting an essentially free affine action on a [Formula: see text]-tree is relatively hyperbolic with nilpotent parabolics, is locally relatively quasiconvex, and has solvable word, conjugacy and isomorphism problems. Conversely, given a graph of groups satisfying certain conditions, we show how an affine action of its fundamental group can be constructed. Specialising to the case of free affine actions, we obtain a large class of groups that have a free affine action on a [Formula: see text]-tree but that do not act freely by isometries on any [Formula: see text]-tree. We also give an example of a group that admits a free isometric action on a [Formula: see text]-tree but which is not residually nilpotent.

scholarly journals AUTOMATIC STRUCTURES AND BOUNDARIES FOR GRAPHS OF GROUPS

1994 ◽  
Vol 04 (04) ◽  
pp. 591-616 ◽  
Author(s):  
WALTER D. NEUMANN ◽  
MICHAEL SHAPIRO
Keyword(s):  
Fundamental Group ◽  
Equivalence Relation ◽  
Infinite Product ◽  
Graph Product ◽  
Graph Of Groups ◽  
Automatic Structures ◽  
Graphs Of Groups ◽  
Underlying Graph ◽  
Edge Group ◽  
Labelled Graphs

We study the synchronous and asynchronous automatic structures on the fundamental group of a graph of groups in which each edge group is finite. Up to a natural equivalence relation, the set of biautomatic structures on such a graph product bijects to the product of the sets of biautomatic structures on the vertex groups. The set of automatic structures is much richer. Indeed, it is dense in the infinite product of the sets of automatic structures of all conjugates of the vertex groups. We classify these structures by a class of labelled graphs which “mimic” the underlying graph of the graph of groups. Analogous statements hold for asynchronous automatic structures. We also discuss the boundaries of these structures.

scholarly journals The Farrell-Jones isomorphism conjecture for 3-manifold groups

2007 ◽  
Vol 1 (1) ◽  
pp. 49-82 ◽  
Author(s):  
S.K. Roushon
Keyword(s):  
Fundamental Group ◽  
Large Class ◽  
Partial Solution ◽  
Fundamental Groups ◽  
Problem List ◽  
Index Subgroup ◽  
Main Aspect ◽  
Graph Of Groups ◽  
Genus 2 ◽  
Finite Index Subgroup

AbstractWe show that the Fibered Isomorphism Conjecture (FIC) of Farrell and Jones corresponding to the stable topological pseudoisotopy functor is true for fundamental groups of a large class of 3-manifolds. We also prove that if the FIC is true for irreducible 3-manifold groups then it is true for all 3-manifold groups. In fact, this follows from a more general result we prove, namely we show that if the FIC is true for each vertex group of a graph of groups with trivial edge groups then the FIC is true for the fundamental group of the graph of groups. This result is part of a program to prove the FIC for the fundamental group of a graph of groups where all the vertex and edge groups satisfy the FIC. A consequence of the first result gives a partial solution to a problem in the problem list of R. Kirby. We also deduce that the FIC is true for a class of virtually P D3-groups.Another main aspect of this article is to prove the FIC for all Haken 3-manifold groups assuming that the FIC is true for B-groups. By definition a B-group contains a finite index subgroup isomorphic to the fundamental group of a compact irreducible 3-manifold with incompressible nonempty boundary so that each boundary component is of genus ≥ 2. We also prove the FIC for a large class of B-groups and moreover, using a recent result of L.E. Jones we show that the surjective part of the FIC is true for any B-group.

scholarly journals Residually finite tubular groups

2019 ◽  
Vol 150 (6) ◽  
pp. 2937-2951
Author(s):  
Nima Hoda ◽  
Daniel T. Wise ◽  
Daniel J. Woodhouse
Keyword(s):  
Finite Graph ◽  
Vertex Group ◽  
Residual Finiteness ◽  
Graph Of Groups ◽  
Residually Finite ◽  
Single Vertex

A tubular group G is a finite graph of groups with ℤ2 vertex groups and ℤ edge groups. We characterize residually finite tubular groups: G is residually finite if and only if its edge groups are separable. Methods are provided to determine if G is residually finite. When G has a single vertex group an algorithm is given to determine residual finiteness.

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