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.

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.

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 Embeddings of quaternion space in S4

1998 ◽  
Vol 65 (3) ◽  
pp. 313-325 ◽  
Author(s):  
Atsuko Katanaga ◽  
Osamu Saeki
Keyword(s):  
Projective Plane ◽  
Fundamental Group ◽  
Euler Number ◽  
Projective Planes ◽  
Connected Components ◽  
Connected Component ◽  
Real Projective Plane ◽  
Quaternion Group ◽  
Topological Embedding ◽  
Quaternion Space

AbstractConsider a (real) projective plane which is topologically locally flatly embedded in S4. It is known that it always admits a 2-disk bundle neighborhood, whose boundary is homeomorphic to the quaternion space Q, the total space of the nonorientable S1-bundle over RP2 with Euler number ± 2, with fundamental group isomorphic to the quaternion group of order eight. Conversely let f: Q → S4 be an arbitrary locally flat topological embedding. Then we show that the closure of each connected component of S4 − f(Q) is always homeomorphic to the exterior of a topologically locally flatly embedded projective plane in S4. We also show that, for a large class of embedded projective planes in S4, a pair of exteriors of such embedded projective planes is always realized as the closures of the connected components of S4 − f(Q) for some locally flat topological embedding f: Q → S4.

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 Simplicial volume of compact manifolds with amenable boundary

2014 ◽  
Vol 07 (01) ◽  
pp. 23-46 ◽  
Author(s):  
Sungwoon Kim ◽  
Thilo Kuessner
Keyword(s):  
Fundamental Group ◽  
Finite Volume ◽  
Compact Manifold ◽  
Riemannian Metric ◽  
Compact Manifolds ◽  
Simplicial Volume ◽  
Curved Manifolds ◽  
Negatively Curved ◽  
Path Component ◽  
Proportionality Principle

Let M be the interior of a connected, oriented, compact manifold V of dimension at least 2. If each path component of ∂V has amenable fundamental group, then we prove that the simplicial volume of M is equal to the relative simplicial volume of V and also to the geometric (Lipschitz) simplicial volume of any Riemannian metric on M whenever the latter is finite. As an application we establish the proportionality principle for the simplicial volume of complete, pinched negatively curved manifolds of finite volume.

scholarly journals The characterization of a line arrangement whose fundamental group of the complement is a direct sum of free groups

2010 ◽  
Vol 10 (3) ◽  
pp. 1285-1304 ◽  
Author(s):  
Meital Eliyahu ◽  
Eran Liberman ◽  
Malka Schaps ◽  
Mina Teicher
Keyword(s):  
Fundamental Group ◽  
Free Groups ◽  
Line Arrangement ◽  
Direct Sum
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