scholarly journals Cohomology groups invariant under continuous orbit equivalence

2020 ◽  
pp. 1-35
Author(s):  
Yongle Jiang
Keyword(s):  
Group Action ◽  
Continuous Group ◽  
Orbit Equivalence ◽  
Cohomology Groups ◽  
Bounded Cohomology ◽  
Homology Groups ◽  
Free Actions ◽  
Uniformly Finite Homology

By the work of Brodzki–Niblo–Nowak–Wright and Monod, topological amenability of a continuous group action can be characterized using uniformly finite homology groups or bounded cohomology groups associated to this action. We show that (certain variations of) these groups are invariants for topologically free actions under continuous orbit equivalence.

Orbit equivalence and actions of

2006 ◽  
Vol 71 (1) ◽  
pp. 265-282 ◽  
Author(s):  
Asge Törnquist
Keyword(s):  
Probability Space ◽  
Free Groups ◽  
Orbit Equivalence ◽  
Borel Probability ◽  
Free Actions ◽  
Measure Preserving Transformations

AbstractIn this paper we show that there are “E0 many” orbit inequivalent free actions of the free groups , 2 ≤ n ≤ ∞ by measure preserving transformations on a standard Borel probability space. In particular, there are uncountably many such actions.

Uniqueness of Free Actions on S3 Respecting a Knot

1987 ◽  
Vol 39 (4) ◽  
pp. 969-982 ◽  
Author(s):  
Michel Boileau ◽  
Erica Flapan
Keyword(s):  
Fixed Point ◽  
Cyclic Group ◽  
Group Action ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Cyclic Group Action ◽  
Periodic Diffeomorphisms ◽  
Free Actions

In this paper we consider free actions of finite cyclic groups on the pair (S3, K), where K is a knot in S3. That is, we look at periodic diffeo-morphisms f of (S3, K) such that fn is fixed point free, for all n less than the order of f. Note that such actions are always orientation preserving. We will show that if K is a non-trivial prime knot then, up to conjugacy, (S3, K) has at most one free finite cyclic group action of a given order. In addition, if all of the companions of K are prime, then all of the free periodic diffeo-morphisms of (S3, K) are conjugate to elements of one cyclic group which acts freely on (S3, K). More specifically, we prove the following two theorems.THEOREM 1. Let K be a non-trivial prime knot. If f and g are free periodic diffeomorphisms of (S3, K) of the same order, then f is conjugate to a power of g.

scholarly journals TRIPLY-GRADED LINK HOMOLOGY AND HOCHSCHILD HOMOLOGY OF SOERGEL BIMODULES

2007 ◽  
Vol 18 (08) ◽  
pp. 869-885 ◽  
Author(s):  
MIKHAIL KHOVANOV
Keyword(s):  
Group Action ◽  
Braid Group ◽  
Polynomial Algebra ◽  
Homology Theory ◽  
Hochschild Homology ◽  
Polynomial Algebras ◽  
Homology Groups ◽  
Link Homology ◽  
Soergel Bimodules ◽  
Braid Group Action

We consider a class of bimodules over polynomial algebras which were originally introduced by Soergel in relation to the Kazhdan–Lusztig theory, and which describe a direct summand of the category of Harish–Chandra modules for sl(n). Rouquier used Soergel bimodules to construct a braid group action on the homotopy category of complexes of modules over a polynomial algebra. We apply Hochschild homology to Rouquier's complexes and produce triply-graded homology groups associated to a braid. These groups turn out to be isomorphic to the groups previously defined by Lev Rozansky and the author, which depend, up to isomorphism and overall shift, only on the closure of the braid. Consequently, our construction produces a homology theory for links.

Groups Acting on Trees

2017 ◽  
Author(s):  
Dan Margalit
Keyword(s):  
Free Group ◽  
Group Action ◽  
Research Projects ◽  
Nontrivial Element ◽  
Connected Graphs ◽  
Farey Tree ◽  
Groups Acting On Trees ◽  
Free Actions

This chapter considers groups acting on trees. It examines which groups act on which spaces and, if a group does act on a space, what it says about the group. These spaces are called trees—that is, connected graphs without cycles. A group action on a tree is free if no nontrivial element of the group preserves any vertex or any edge of the tree. The chapter first presents the theorem stating that: If a group G acts freely on a tree, then G is a free group. The condition that G is free is equivalent to the condition that G acts freely on a tree. The discussion then turns to the Farey tree and shows how to construct the Farey complex using the Farey graph. The chapter concludes by describing free and non-free actions on trees. Exercises and research projects are included.

scholarly journals Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions

2015 ◽  
Vol 36 (5) ◽  
pp. 1557-1581 ◽  
Author(s):  
KENGO MATSUMOTO ◽  
HIROKI MATUI
Keyword(s):  
Zeta Functions ◽  
Periodic Points ◽  
Orbit Equivalence ◽  
Cohomology Groups ◽  
Borel Measures ◽  
Orbit Equivalent ◽  
Positive Elements ◽  
Markov Shifts ◽  
Sigma B

For continuously orbit equivalent one-sided topological Markov shifts $(X_{A},{\it\sigma}_{A})$ and $(X_{B},{\it\sigma}_{B})$, their eventually periodic points and cocycle functions are studied. As a result, we directly construct an isomorphism between their ordered cohomology groups $(\bar{H}^{A},\bar{H}_{+}^{A})$ and $(\bar{H}^{B},\bar{H}_{+}^{B})$. We also show that the cocycle functions for the continuous orbit equivalences give rise to positive elements of their ordered cohomology groups, so that the zeta functions of continuously orbit equivalent topological Markov shifts are related. The set of Borel measures is shown to be invariant under continuous orbit equivalence of one-sided topological Markov shifts.

COMPUTATION OF LOW-DIMENSIONAL (CO)HOMOLOGY GROUPS FOR INFINITE SEQUENCES OF p-GROUPS WITH FIXED COCLASS

2011 ◽  
Vol 21 (04) ◽  
pp. 635-649 ◽  
Author(s):  
BETTINA EICK ◽  
DÖRTE FEICHTENSCHLAGER
Keyword(s):  
Infinite Sequence ◽  
Cohomology Groups ◽  
Homology Groups ◽  
Low Dimensional ◽  
Infinite Sequences

Eick and Leedham-Green introduced infinite sequences of p-groups of fixed coclass. Here we describe an algorithm to compute the Schur multiplicators of all groups in an infinite sequence simultaneously. Based on this, we prove that these Schur multiplicators can be described by a single parametrised presentation; this confirms a conjecture by Eick. Similar results are obtained for certain low-dimensional cohomology groups; these results support a conjecture by Carlson.

Homology and Cohomology of the Super Schrödinger Algebra S(1/1) with Coefficients in the Trivial Module

2019 ◽  
Vol 26 (04) ◽  
pp. 615-628
Author(s):  
Yan He ◽  
Yuezhu Wu ◽  
Linsheng Zhu
Keyword(s):  
Cohomology Groups ◽  
Homology Groups ◽  
Dimensional Spacetime ◽  
Trivial Module ◽  
Schrödinger Algebra

In this paper we study the homology and cohomology groups of the super Schrödinger algebra [Formula: see text] in (1 + 1)-dimensional spacetime. We explicitly compute the homology groups of [Formula: see text] with coefficients in the trivial module. Then using duality, we finally obtain the dimensions of the cohomology groups of [Formula: see text] with coefficients in the trivial module.

scholarly journals RELATIONSHIP BETWEEN EQUIVARIANT GERBES AND GERBES OVER THE QUOTIENT SPACE

2005 ◽  
Vol 07 (02) ◽  
pp. 207-226 ◽  
Author(s):  
KIYONORI GOMI
Keyword(s):  
Group Action ◽  
Lie Group ◽  
Quotient Space ◽  
Cohomology Groups ◽  
Lie Group Action

By means of cohomology groups, we study relationships between equivariant gerbes with connection over a manifold with a Lie group action and gerbes with connection over the quotient space.

scholarly journals Excision in Banach simplicial and cyclic cohomology

1998 ◽  
Vol 41 (2) ◽  
pp. 411-427 ◽  
Author(s):  
Zinaida A. Lykova
Keyword(s):  
Exact Sequence ◽  
Banach Algebras ◽  
Approximate Identity ◽  
Cyclic Cohomology ◽  
Cohomology Groups ◽  
Continuous Version ◽  
Homology Groups ◽  
Exact Sequences

We prove that, for every extension of Banach algebras 0 → B →A → D → 0 such that B has a left or right bounded approximate identity, the existence of an associated long exact sequence of Banach simplicial or cyclic cohomology groups is equivalent to the existence of one for homology groups. It follows from the continuous version of a result of Wodzicki that associated long exact sequences exist. In particular, they exist for every extension of C*-algebras.

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