Archimedean actions on median pretrees

2001 ◽  
Vol 130 (3) ◽  
pp. 383-400 ◽  
Author(s):  
BRIAN H. BOWDITCH ◽  
JOHN CRISP
Keyword(s):  
Group Actions ◽  
General Setting ◽  
Countable Group ◽  
Convergence Group ◽  
Isometric Actions ◽  
Betweenness Relations

In this paper we consider group actions on generalized treelike structures (termed ‘pretrees’) defined simply in terms of betweenness relations. Using a result of Levitt, we show that if a countable group admits an archimedean action on a median pretree, then it admits an action by isometries on an ℝ-tree. Thus the theory of isometric actions on ℝ-trees may be extended to a more general setting where it merges naturally with the theory of right-orderable groups. This approach has application also to the study of convergence group actions on continua.

scholarly journals On laminar groups, Tits alternatives and convergence group actions on 𝑆2

2019 ◽  
Vol 22 (3) ◽  
pp. 359-381
Author(s):  
Juan Alonso ◽  
Hyungryul Baik ◽  
Eric Samperton
Keyword(s):  
Group Actions ◽  
Convergence Group ◽  
Tits Alternative ◽  
Fibered Group ◽  
Torsion Free

Abstract Following previous work of the second author, we establish more properties of groups of circle homeomorphisms which admit invariant laminations. In this paper, we focus on a certain type of such groups, so-called pseudo-fibered groups, and show that many 3-manifold groups are examples of pseudo-fibered groups. We then prove that torsion-free pseudo-fibered groups satisfy a Tits alternative. We conclude by proving that a purely hyperbolic pseudo-fibered group acts on the 2-sphere as a convergence group. This leads to an interesting question if there are examples of pseudo-fibered groups other than 3-manifold groups.

scholarly journals Amenability, Kazhdan's property T, strong ergodicity and invariant means for ergodic group-actions

1981 ◽  
Vol 1 (2) ◽  
pp. 223-236 ◽  
Author(s):  
Klaus Schmidt
Keyword(s):  
Group Action ◽  
Measure Space ◽  
Group Actions ◽  
Finite Measure ◽  
Countable Group ◽  
Invariant Sets ◽  
Amenable Groups ◽  
Strong Ergodicity ◽  
Property T ◽  
Invariant Means

AbstractThis paper discusses the relations between the following properties o finite measure preserving ergodic actions of a countable group G: strong ergodicity (i.e. the non-existence of almost invariant sets), uniqueness of G-invariant means on the measure space carrying the group action, and certain cohomological properties. Using these properties one can characterize all actions of amenable groups and of groups with Kazhdan's property T. For groups which fall in between these two definations these notions lead to some interesting examples.

scholarly journals Examples in the entropy theory of countable group actions

2019 ◽  
Vol 40 (10) ◽  
pp. 2593-2680 ◽  
Author(s):  
LEWIS BOWEN
Keyword(s):  
Classical Theory ◽  
Group Actions ◽  
Countable Group ◽  
Classification Theory ◽  
Entropy Theory ◽  
Survey Article ◽  
Sofic Entropy ◽  
Factor Maps ◽  
Measure Preserving Actions ◽  
Classical Entropy

Kolmogorov–Sinai entropy is an invariant of measure-preserving actions of the group of integers that is central to classification theory. There are two recently developed invariants, sofic entropy and Rokhlin entropy, that generalize classical entropy to actions of countable groups. These new theories have counterintuitive properties such as factor maps that increase entropy. This survey article focusses on examples, many of which have not appeared before, that highlight the differences and similarities with classical theory.

COMPUTABLE POLISH GROUP ACTIONS

2018 ◽  
Vol 83 (2) ◽  
pp. 443-460
Author(s):  
ALEXANDER MELNIKOV ◽  
ANTONIO MONTALBÁN
Keyword(s):  
Set Theory ◽  
Group Actions ◽  
General Setting ◽  
Structure Theory ◽  
Computable Structure ◽  
Descriptive Set Theory ◽  
Sufficient Condition ◽  
Computable Structure Theory ◽  
Image Position ◽  
Descriptive Set

AbstractUsing methods from computable analysis, we establish a new connection between two seemingly distant areas of logic: computable structure theory and invariant descriptive set theory. We extend several fundamental results of computable structure theory to the more general setting of topological group actions. As we will see, the usual action of ${S_\infty }$ on the space of structures in a given language is effective in a certain algorithmic sense that we need, and ${S_\infty }$ itself carries a natural computability structure (to be defined). Among other results, we give a sufficient condition for an orbit under effective ${\cal G}$-action of a computable Polish ${\cal G}$ to split into infinitely many disjoint effective orbits. Our results are not only more general than the respective results in computable structure theory, but they also tend to have proofs different from (and sometimes simpler than) the previously known proofs of the respective prototype results.

scholarly journals On nonmeasurable selectors of countable group actions

2009 ◽  
Vol 202 (3) ◽  
pp. 281-294 ◽  
Author(s):  
Piotr Zakrzewski
Keyword(s):  
Group Actions ◽  
Countable Group

Recurrence in mean for group actions

2019 ◽  
Vol 20 (01) ◽  
pp. 2050006
Author(s):  
Cao Zhao ◽  
Yong Ji
Keyword(s):  
Hausdorff Measure ◽  
Metric Space ◽  
Group Actions ◽  
Finite Measure ◽  
Countable Group ◽  
General Group ◽  
Mean Values ◽  
Measurable Set ◽  
The Mean

In this paper, the mean values of the recurrence are computed for general group actions. Let [Formula: see text] be a metric space with a finite measure [Formula: see text] and [Formula: see text] be a countable group acting on [Formula: see text]. Let [Formula: see text] be a sequence of subsets of [Formula: see text] with [Formula: see text] and put [Formula: see text]. If the Hausdorff measure [Formula: see text] is finite on [Formula: see text] and [Formula: see text] is [Formula: see text]-invariant. We assume that [Formula: see text] and [Formula: see text] are concordant. Then the function [Formula: see text] is [Formula: see text]-integrable and for any [Formula: see text]-measurable set [Formula: see text] we have [Formula: see text] If moreover, [Formula: see text] then [Formula: see text] without the concordance condition for the measure [Formula: see text] and [Formula: see text]

scholarly journals Directed unions of local monoidal transforms and GCD domains

2019 ◽  
Vol 19 (04) ◽  
pp. 2050061
Author(s):  
Lorenzo Guerrieri
Keyword(s):  
Prime Ideal ◽  
Local Ring ◽  
Maximal Ideal ◽  
General Setting ◽  
Regular Local Ring ◽  
Dimension Formula

Let [Formula: see text] be a regular local ring of dimension [Formula: see text]. A local monoidal transform of [Formula: see text] is a ring of the form [Formula: see text], where [Formula: see text] is a regular parameter, [Formula: see text] is a regular prime ideal of [Formula: see text] and [Formula: see text] is a maximal ideal of [Formula: see text] lying over [Formula: see text] In this paper, we study some features of the rings [Formula: see text] obtained as infinite directed union of iterated local monoidal transforms of [Formula: see text]. In order to study when these rings are GCD domains, we also provide results in the more general setting of directed unions of GCD domains.

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