Marked length rigidity for one-dimensional spaces

In a compact geodesic metric space of topological dimension one, the minimal length of a loop in a free homotopy class is well-defined, and provides a function [Formula: see text] (the value [Formula: see text] being assigned to loops which are not freely homotopic to any rectifiable loops). This function is the marked length spectrum. We introduce a subset [Formula: see text], which is the union of all non-constant minimal loops of finite length. We show that if [Formula: see text] is a compact, non-contractible, geodesic space of topological dimension one, then [Formula: see text] deformation retracts to [Formula: see text]. Moreover, [Formula: see text] can be characterized as the minimal subset of [Formula: see text] to which [Formula: see text] deformation retracts. Let [Formula: see text] be a pair of compact, non-contractible, geodesic metric spaces of topological dimension one, and set [Formula: see text]. We prove that any isomorphism [Formula: see text] satisfying [Formula: see text], forces the existence of an isometry [Formula: see text] which induces the map [Formula: see text] on the level of fundamental groups. Thus, for compact, non-contractible, geodesic spaces of topological dimension one, the marked length spectrum completely determines the subset [Formula: see text] up to isometry.

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A CRITERION FOR ERDŐS SPACES

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091504000823 ◽

2005 ◽

Vol 48 (3) ◽

pp. 595-601 ◽

Cited By ~ 10

Author(s):

Jan J. Dijkstra

Keyword(s):

Hilbert Space ◽

Banach Spaces ◽

Real Hilbert Space ◽

Topological Dimension ◽

One Dimensional ◽

The Real ◽

General Space ◽

Dimension One ◽

Totally Disconnected

AbstractIn 1940 Paul Erdős introduced the ‘rational Hilbert space’, which consists of all vectors in the real Hilbert space $\ell^2$ that have only rational coordinates. He showed that this space has topological dimension one, yet it is totally disconnected and homeomorphic to its square. In this note we generalize the construction of this peculiar space and we consider all subspaces $\mathcal{E}$ of the Banach spaces $\ell^p$ that are constructed as ‘products’ of zero-dimensional subsets $E_n$ of $\mathbb{R}$. We present an easily applied criterion for deciding whether a general space of this type is one dimensional. As an application we find that if such an $\mathcal{E}$ is closed in $\ell^p$, then it is homeomorphic to complete Erdős space if and only if $\dim\mathcal{E}>0$ and every $E_n$ is zero dimensional.

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On geometric polygroups

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2020-0002 ◽

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.

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Geometric characterizations of virtually free groups

Journal of Algebra and Its Applications ◽

10.1142/s0219498817501808 ◽

2016 ◽

Vol 16 (09) ◽

pp. 1750180 ◽

Cited By ~ 1

Author(s):

Vítor Araújo ◽

Pedro V. Silva

Keyword(s):

Cayley Graph ◽

Metric Space ◽

Metric Spaces ◽

Free Groups ◽

Finitely Generated ◽

Finitely Generated Group ◽

Geometric Conditions ◽

Geodesic Metric Space ◽

Geodesic Metric ◽

Gromov Product

Four geometric conditions on a geodesic metric space, which are stronger variants of classical conditions characterizing hyperbolicity (featuring [Formula: see text]-thin polygons, the Gromov product or the mesh of triangles), are proved to be equivalent. They define the class of polygon hyperbolic geodesic metric spaces. In the particular case of the Cayley graph of a finitely generated group, it is shown that they characterize virtually free groups.

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DIAGONALIZATION MODULO NORM IDEALS AND HAUSDORFF DIMENSIONALITY

International Journal of Mathematics ◽

10.1142/s0129167x00000520 ◽

2000 ◽

Vol 11 (08) ◽

pp. 1057-1078

Author(s):

JINGBO XIA

Keyword(s):

Hausdorff Measure ◽

Metric Spaces ◽

Adjoint Operator ◽

Trace Class ◽

Multiplication Operators ◽

Von Neumann ◽

Dimensional Hausdorff Measure ◽

One Dimensional ◽

Compact Metric Spaces ◽

The One

Kuroda's version of the Weyl-von Neumann theorem asserts that, given any norm ideal [Formula: see text] not contained in the trace class [Formula: see text], every self-adjoint operator A admits the decomposition A=D+K, where D is a self-adjoint diagonal operator and [Formula: see text]. We extend this theorem to the setting of multiplication operators on compact metric spaces (X, d). We show that if μ is a regular Borel measure on X which has a σ-finite one-dimensional Hausdorff measure, then the family {Mf:f∈ Lip (X)} of multiplication operators on T2(X, μ) can be simultaneously diagonalized modulo any [Formula: see text]. Because the condition [Formula: see text] in general cannot be dropped (Kato-Rosenblum theorem), this establishes a special relation between [Formula: see text] and the one-dimensional Hausdorff measure. The main result of the paper is that such a relation breaks down in Hausdorff dimensions p>1.

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A Note on One-dimensional Varieties Over the Complex p-adic Field

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v11i4.3281 ◽

2018 ◽

Vol 11 (4) ◽

pp. 1046-1057

Author(s):

Amran Dalloul

Keyword(s):

One Dimensional ◽

Dimension One

In this paper, we study the varieties V ⊆ C 4 p of dimension one that contain points of the form (x1, x2, exp(x1), exp(x2)) by using tools from Non-Archimedian Analysis.

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Cogrowth for group actions with strongly contracting elements

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.123 ◽

2018 ◽

Vol 40 (7) ◽

pp. 1738-1754 ◽

Cited By ~ 1

Author(s):

GOULNARA N. ARZHANTSEVA ◽

CHRISTOPHER H. CASHEN

Keyword(s):

Mapping Class Groups ◽

Hyperbolic Spaces ◽

Hyperbolic Groups ◽

Mapping Class ◽

Small Cancellation ◽

Class Groups ◽

Original Result ◽

Geodesic Metric Space ◽

Geodesic Metric ◽

Right Angled Artin Groups

Let $G$ be a group acting properly by isometries and with a strongly contracting element on a geodesic metric space. Let $N$ be an infinite normal subgroup of $G$ and let $\unicode[STIX]{x1D6FF}_{N}$ and $\unicode[STIX]{x1D6FF}_{G}$ be the growth rates of $N$ and $G$ with respect to the pseudo-metric induced by the action. We prove that if $G$ has purely exponential growth with respect to the pseudo-metric, then $\unicode[STIX]{x1D6FF}_{N}/\unicode[STIX]{x1D6FF}_{G}>1/2$. Our result applies to suitable actions of hyperbolic groups, right-angled Artin groups and other CAT(0) groups, mapping class groups, snowflake groups, small cancellation groups, etc. This extends Grigorchuk’s original result on free groups with respect to a word metric and a recent result of Matsuzaki, Yabuki and Jaerisch on groups acting on hyperbolic spaces to a much wider class of groups acting on spaces that are not necessarily hyperbolic.

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Marked length rigidity for Fuchsian buildings

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2018.12 ◽

2018 ◽

Vol 39 (12) ◽

pp. 3262-3291

Author(s):

DAVID CONSTANTINE ◽

JEAN-FRANÇOIS LAFONT

Keyword(s):

Automorphism Group ◽

Riemannian Metrics ◽

Length Spectrum ◽

Negatively Curved ◽

Spectrum Function ◽

Marked Length Spectrum

We consider finite $2$-complexes $X$ that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT($-1$) metrics on $X$, which are piecewise hyperbolic and satisfy a natural non-singularity condition at vertices, are marked length spectrum rigid within certain classes of negatively curved, piecewise Riemannian metrics on $X$. As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of $X$.

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Best proximity point results in geodesic metric spaces

Fixed Point Theory and Applications ◽

10.1186/1687-1812-2012-234 ◽

2012 ◽

Vol 2012 (1) ◽

pp. 234 ◽

Cited By ~ 4

Author(s):

Maryam A Alghamdi ◽

Mohammed A Alghamdi ◽

Naseer Shahzad

Keyword(s):

Metric Spaces ◽

Best Proximity Point ◽

Geodesic Metric

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CHAOS ON ONE-DIMENSIONAL COMPACT METRIC SPACES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127412502598 ◽

2012 ◽

Vol 22 (10) ◽

pp. 1250259 ◽

Cited By ~ 8

Author(s):

ZDENĚK KOČAN

Keyword(s):

Topological Entropy ◽

Metric Spaces ◽

Chaotic Behavior ◽

Homoclinic Trajectory ◽

One Dimensional ◽

Distributional Chaos ◽

Compact Metric Spaces ◽

Continuous Maps

We consider various kinds of chaotic behavior of continuous maps on compact metric spaces: the positivity of topological entropy, the existence of a horseshoe, the existence of a homoclinic trajectory (or perhaps, an eventually periodic homoclinic trajectory), three levels of Li–Yorke chaos, three levels of ω-chaos and distributional chaos of type 1. The relations between these properties are known when the space is an interval. We survey the known results in the case of trees, graphs and dendrites.

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