scholarly journals Recurrent sets and shadowing for finitely generated semigroup actions on metric spaces

2021 ◽  
pp. 1-15
Author(s):  
Xinxing WU
Keyword(s):  
Metric Spaces ◽  
Finitely Generated ◽  
Semigroup Actions

scholarly journals COARSE COHERENCE OF METRIC SPACES AND GROUPS AND ITS PERMANENCE PROPERTIES

2018 ◽  
Vol 98 (3) ◽  
pp. 422-433
Author(s):  
BORIS GOLDFARB ◽  
JONATHAN L. GROSSMAN
Keyword(s):  
Metric Spaces ◽  
General Context ◽  
Metric Geometry ◽  
Finitely Generated ◽  
Coherence Property ◽  
Finitely Generated Groups ◽  
Permanence Property ◽  
Permanence Properties ◽  
New Framework

We introduce properties of metric spaces and, specifically, finitely generated groups with word metrics, which we call coarse coherence and coarse regular coherence. They are geometric counterparts of the classical algebraic notion of coherence and the regular coherence property of groups defined and studied by Waldhausen. The new properties can be defined in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. The new framework allows us to prove structural results by developing permanence properties, including the particularly important fibering permanence property, for coarse regular coherence.

Finitely approximate groups and actions Part I: The Ribes–Zalesskiĭ property

2011 ◽  
Vol 76 (4) ◽  
pp. 1297-1306 ◽  
Author(s):  
Christian Rosendal
Keyword(s):  
Metric Space ◽  
Discrete Group ◽  
Metric Spaces ◽  
Finitely Generated ◽  
Partial Isometries ◽  
Profinite Topology

AbstractWe investigate extensions of S. Solecki's theorem on closing off finite partial isometries of metric spaces [11] and obtain the following exact equivalence: any action of a discrete group Γ by isometries of a metric space is finitely approximable if and only if any product of finitely generated subgroups of Γ is closed in the profinite topology on Γ.

scholarly journals On geometric polygroups

2020 ◽  
Vol 28 (1) ◽  
pp. 17-33
Author(s):  
F. Arabpur ◽  
M. Jafarpour ◽  
M. Aminizadeh ◽  
S. Hoskova-Mayerova
Keyword(s):  
Cayley Graph ◽  
Metric Space ◽  
Metric Spaces ◽  
Finitely Generated ◽  
Geodesic Metric Space ◽  
Geodesic Metric

AbstractIn this paper, we introduce a geodesic metric space called generalized Cayley graph (gCay(P,S)) on a finitely generated polygroup. We define a hyperaction of polygroup on gCayley graph and give some properties of this hyperaction. We show that gCayley graphs of a polygroup by two different generators are quasi-isometric. Finally, we express a connection between finitely generated polygroups and geodesic metric spaces.

Hierarchy for groups acting on hyperbolic ℤn-spaces

2021 ◽  
pp. 1-28
Author(s):  
Andrei-Paul Grecianu ◽  
Alexei Myasnikov ◽  
Denis Serbin
Keyword(s):  
Abelian Group ◽  
Systematic Study ◽  
Metric Spaces ◽  
Free Groups ◽  
Lexicographic Order ◽  
Finitely Generated ◽  
Princeton University ◽  
Finitely Generated Groups ◽  
The One ◽  
The Right

In [A.-P. Grecianu, A. Kvaschuk, A. G. Myasnikov and D. Serbin, Groups acting on hyperbolic [Formula: see text]-metric spaces, Int. J. Algebra Comput. 25(6) (2015) 977–1042], the authors initiated a systematic study of hyperbolic [Formula: see text]-metric spaces, where [Formula: see text] is an ordered abelian group, and groups acting on such spaces. The present paper concentrates on the case [Formula: see text] taken with the right lexicographic order and studies the structure of finitely generated groups acting on hyperbolic [Formula: see text]-metric spaces. Under certain constraints, the structure of such groups is described in terms of a hierarchy (see [D. T. Wise, The Structure of Groups with a Quasiconvex Hierarchy[Formula: see text][Formula: see text]AMS-[Formula: see text], Annals of Mathematics Studies (Princeton University Press, 2021)]) similar to the one established for [Formula: see text]-free groups in [O. Kharlampovich, A. G. Myasnikov, V. N. Remeslennikov and D. Serbin, Groups with free regular length functions in [Formula: see text], Trans. Amer. Math. Soc. 364 (2012) 2847–2882].

scholarly journals A bridge theorem for the entropy of semigroup actions

2020 ◽  
Vol 8 (1) ◽  
pp. 46-57
Author(s):  
Anna Giordano Bruno
Keyword(s):  
Abelian Group ◽  
Topological Entropy ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Finitely Generated ◽  
Semigroup Action ◽  
Locally Compact ◽  
Semigroup Actions ◽  
Dual Action ◽  
Totally Disconnected

AbstractThe topological entropy of a semigroup action on a totally disconnected locally compact abelian group coincides with the algebraic entropy of the dual action. This relation holds both for the entropy relative to a net and for the receptive entropy of finitely generated monoid actions.

A QUASI-ISOMETRY INVARIANT LOOP SHORTENING PROPERTY FOR GROUPS

2008 ◽  
Vol 18 (08) ◽  
pp. 1243-1257 ◽  
Author(s):  
STEPHEN G. BRICK ◽  
JON M. CORSON ◽  
DOHYOUNG RYANG
Keyword(s):  
Isoperimetric Inequality ◽  
Metric Spaces ◽  
Cayley Graphs ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Quadratic Isoperimetric Inequality ◽  
Finitely Presented

We first introduce a loop shortening property for metric spaces, generalizing the property considered by M. Elder on Cayley graphs of finitely generated groups. Then using this metric property, we define a very broad loop shortening property for finitely generated groups. Our definition includes Elder's groups, and unlike his definition, our property is obviously a quasi-isometry invariant of the group. Furthermore, all finitely generated groups satisfying this general loop shortening property are also finitely presented and satisfy a quadratic isoperimetric inequality. Every CAT(0) cubical group is shown to have this general loop shortening property.

scholarly journals An intermediate quasi-isometric invariant between subexponential asymptotic dimension growth and Yu's property A

2014 ◽  
Vol 24 (06) ◽  
pp. 909-922 ◽  
Author(s):  
Izhar Oppenheim
Keyword(s):  
Metric Spaces ◽  
Asymptotic Dimension ◽  
Finitely Generated ◽  
Property A ◽  
Large Depth ◽  
Finitely Generated Groups

We present a new quasi-isometric invariant for metric spaces that we name asymptotically large depth. For finitely generated groups we show that this invariant implies Yu's property A and is implied by subexponential asymptotic dimension growth.

scholarly journals Semigroup Actions on Intuitionistic Fuzzy Metric Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-5
Author(s):  
Yaoyao Lan ◽  
Qingguo Li
Keyword(s):  
Dynamical Systems ◽  
Topological Semigroup ◽  
Metric Spaces ◽  
Topological Transitivity ◽  
Fuzzy Metric Spaces ◽  
Semigroup Actions ◽  
Fuzzy Metric ◽  
Intuitionistic Fuzzy ◽  
Dynamical Properties

This paper investigates the dynamical systems in the context of topological semigroup actions on intuitionistic fuzzy metric spaces. We give some concepts such as topological transitivity, point transitivity, and densely point transitivity for such dynamical systems. Particularly, we consider the implications of nonsensitivity and its relation to dynamical properties such as transitivity and equicontinuity.

EMBEDDABILITY OF GENERALISED WREATH PRODUCTS

2014 ◽  
Vol 91 (2) ◽  
pp. 250-263 ◽  
Author(s):  
CHRIS CAVE ◽  
DENNIS DREESEN
Keyword(s):  
Hilbert Space ◽  
Wreath Product ◽  
Metric Spaces ◽  
Wreath Products ◽  
Finitely Generated ◽  
Product Construction ◽  
Finitely Generated Groups

AbstractGiven two finitely generated groups that coarsely embed into a Hilbert space, it is known that their wreath product also embeds coarsely into a Hilbert space. We introduce a wreath product construction for general metric spaces $X,Y,Z$ and derive a condition, called the (${\it\delta}$-polynomial) path lifting property, such that coarse embeddability of $X,Y$ and $Z$ implies coarse embeddability of $X\wr _{Z}Y$. We also give bounds on the compression of $X\wr _{Z}Y$ in terms of ${\it\delta}$ and the compressions of $X,Y$ and $Z$.

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