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 Groups generated by elements with rational fixed points

1997 ◽  
Vol 40 (1) ◽  
pp. 19-30 ◽  
Author(s):  
A. W. Mason
Keyword(s):  
Normal Subgroup ◽  
Fixed Points ◽  
Large Class ◽  
Integral Domain ◽  
Structure Theorem ◽  
Quotient Field ◽  
Linear Fractional Transformations ◽  
Krull Domain ◽  
Dedekind Domains ◽  
Precise Version

Let R be a commutative integral domain and let S be its quotient field. The group GL2(R) acts on Ŝ = S ∪ {∞} as a group of linear fractional transformations in the usual way. Let F2(R, z) be the stabilizer of z ∈ Ŝ in GL2(R) and let F2(R) be the subgroup generated by all F2(R, z). Among the subgroups contained in F2(R) are U2(R), the subgroup generated by all unipotent matrices, and NE2(R), the normal subgroup generated by all elementary matrices.We prove a structure theorem for F2(R, z), when R is a Krull domain. A more precise version holds when R is a Dedekind domain. For a large class of arithmetic Dedekind domains it is known that the groups NE2(R),U2(R) and SL2(R) coincide. An example is given for which all these subgroups are distinct.

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 The fundamental group of a quotient of a product of curves

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Thomas Dedieu ◽  
Fabio Perroni
Keyword(s):  
Fundamental Group ◽  
Structure Theorem

Abstract.We prove a structure theorem for the fundamental group of the quotient

scholarly journals Families of SU(2) representations for mapping cylinders of periodic monodromy

1997 ◽  
Vol 40 (2) ◽  
pp. 383-392
Author(s):  
G. Daskalopoulos ◽  
S. Dostoglou ◽  
R. Wentworth
Keyword(s):  
Fundamental Group ◽  
Fixed Points ◽  
Conjugacy Classes ◽  
Oriented Surface ◽  
3 Dimensional ◽  
Dimensional Mapping

We examine the action of diffeomorphisms of an oriented surface with boundary on the space of conjugacy classes of SU(2) representations of the fundamental group and prove that in the case of a single periodic diffeomorphism the induced action always has fixed points. For the corresponding 3-dimensional mapping cylinders we obtain families of representations parametrized by their value on the longitude of the torus boundary.

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.

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.

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