Geometric characterizations of virtually free groups

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.

Download Full-text

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.

Download Full-text

Quasi-Isometries Need Not Induce Homeomorphisms of Contracting Boundaries with the Gromov Product Topology

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2016-0011 ◽

2016 ◽

Vol 4 (1) ◽

Author(s):

Christopher H. Cashen

Keyword(s):

Hyperbolic Space ◽

Metric Space ◽

Equivalence Classes ◽

Product Topology ◽

Geodesic Metric Space ◽

Geodesic Metric ◽

Gromov Product ◽

Geodesic Rays

AbstractWe consider a ‘contracting boundary’ of a proper geodesic metric space consisting of equivalence classes of geodesic rays that behave like geodesics in a hyperbolic space.We topologize this set via the Gromov product, in analogy to the topology of the boundary of a hyperbolic space. We show that when the space is not hyperbolic, quasi-isometries do not necessarily give homeomorphisms of this boundary. Continuity can fail even when the spaces are required to be CAT(0). We show this by constructing an explicit example.

Download Full-text

COUNTING CONJUGACY CLASSES IN

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972718000047 ◽

2018 ◽

Vol 97 (3) ◽

pp. 412-421

Author(s):

MICHAEL HULL ◽

ILYA KAPOVICH

Keyword(s):

Cayley Graph ◽

Metric Space ◽

Hyperbolic Metric ◽

Conjugacy Classes ◽

Finitely Generated ◽

Finitely Generated Group ◽

Irreducible Elements ◽

Hyperbolic Metric Space

We show that if a finitely generated group$G$has a nonelementary WPD action on a hyperbolic metric space$X$, then the number of$G$-conjugacy classes of$X$-loxodromic elements of$G$coming from a ball of radius$R$in the Cayley graph of$G$grows exponentially in$R$. As an application we prove that for$N\geq 3$the number of distinct$\text{Out}(F_{N})$-conjugacy classes of fully irreducible elements$\unicode[STIX]{x1D719}$from an$R$-ball in the Cayley graph of$\text{Out}(F_{N})$with$\log \unicode[STIX]{x1D706}(\unicode[STIX]{x1D719})$of the order of$R$grows exponentially in$R$.

Download Full-text

ISOPERIMETRIC FUNCTIONS OF GROUPS ACTING ON Lδ-SPACES

Glasgow Mathematical Journal ◽

10.1017/s0017089507003382 ◽

2007 ◽

Vol 49 (1) ◽

pp. 23-28

Author(s):

JON CORSON ◽

DOHYOUNG RYANG

Keyword(s):

Metric Space ◽

Finitely Generated ◽

Isoperimetric Function ◽

Finitely Generated Group ◽

Isoperimetric Functions ◽

Finitely Presented

Abstract.A finitely generated group acting properly, cocompactly, and by isometries on an Lδ-metric space is finitely presented and has a sub-cubic isoperimetric function.

Download Full-text

Mathematical Properties of the Hyperbolicity of Circulant Networks

Advances in Mathematical Physics ◽

10.1155/2015/723451 ◽

2015 ◽

Vol 2015 ◽

pp. 1-11 ◽

Cited By ~ 4

Author(s):

Juan C. Hernández ◽

José M. Rodríguez ◽

José M. Sigarreta

Keyword(s):

Cyclic Group ◽

Metric Space ◽

Geodesic Triangle ◽

Group Of Automorphisms ◽

Mathematical Properties ◽

Geodesic Metric Space ◽

Geodesic Metric ◽

Circulant Networks ◽

Hyperbolicity Constant

IfXis a geodesic metric space andx1,x2,x3∈X, ageodesic triangle T={x1,x2,x3}is the union of the three geodesics[x1x2],[x2x3], and[x3x1]inX. The spaceXisδ-hyperbolic(in the Gromov sense) if any side ofTis contained in aδ-neighborhood of the union of the two other sides, for every geodesic triangleTinX. The study of the hyperbolicity constant in networks is usually a very difficult task; therefore, it is interesting to find bounds for particular classes of graphs. A network is circulant if it has a cyclic group of automorphisms that includes an automorphism taking any vertex to any other vertex. In this paper we obtain several sharp inequalities for the hyperbolicity constant of circulant networks; in some cases we characterize the graphs for which the equality is attained.

Download Full-text

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

Journal of Symbolic Logic ◽

10.2178/jsl/1318338850 ◽

2011 ◽

Vol 76 (4) ◽

pp. 1297-1306 ◽

Cited By ~ 5

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.

Download Full-text

The Herzog–Schönheim conjecture for finitely generated groups

International Journal of Algebra and Computation ◽

10.1142/s0218196719500425 ◽

2019 ◽

Vol 29 (06) ◽

pp. 1083-1112 ◽

Cited By ~ 1

Author(s):

Fabienne Chouraqui

Keyword(s):

Metric Space ◽

Finite Rank ◽

Free Groups ◽

Sufficient Conditions ◽

Combinatorial Approach ◽

Finitely Generated ◽

Covering Spaces ◽

Finitely Generated Groups

Let [Formula: see text] be a group and [Formula: see text] be subgroups of [Formula: see text] of indices [Formula: see text], respectively. In 1974, Herzog and Schönheim conjectured that if [Formula: see text], [Formula: see text], is a coset partition of [Formula: see text], then [Formula: see text] cannot be distinct. We consider the Herzog–Schönheim conjecture for free groups of finite rank and develop a new combinatorial approach, using covering spaces. We define [Formula: see text] the space of coset partitions of [Formula: see text] and show [Formula: see text] is a metric space with interesting properties. We give some sufficient conditions on the coset partition that ensure the conjecture is satisfied and moreover has a neighborhood [Formula: see text] in [Formula: see text] such that all the partitions in [Formula: see text] satisfy also the conjecture.

Download Full-text

Tauberian Conditions for Almost Convergence in a Geodesic Metric Space

Results in Mathematics ◽

10.1007/s00025-020-1169-6 ◽

2020 ◽

Vol 75 (1) ◽

Author(s):

Hadi Khatibzadeh ◽

Hadi Pouladi

Keyword(s):

Metric Space ◽

Almost Convergence ◽

Tauberian Conditions ◽

Geodesic Metric Space ◽

Geodesic Metric

Download Full-text

Growth tightness for groups with contracting elements

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004114000322 ◽

2014 ◽

Vol 157 (2) ◽

pp. 297-319 ◽

Cited By ~ 6

Author(s):

WEN-YUAN YANG

Keyword(s):

Metric Space ◽

Hyperbolic Group ◽

Interesting Consequence ◽

Geodesic Metric Space ◽

Geodesic Metric ◽

Relatively Hyperbolic Group ◽

Floyd Boundary ◽

Relatively Hyperbolic

AbstractWe establish growth tightness for a class of groups acting geometrically on a geodesic metric space and containing a contracting element. As a consequence, any group with non-trivial Floyd boundary are proven to be growth tight with respect to word metrics. In particular, all non-elementary relatively hyperbolic group are growth tight. This generalizes previous works of Arzhantseva-Lysenok and Sambusetti. Another interesting consequence is that CAT(0) groups with rank-1 elements are growth tight with respect to CAT(0)-metric.

Download Full-text

Distortion of the Hyperbolicity Constant of a Graph

The Electronic Journal of Combinatorics ◽

10.37236/2175 ◽

2012 ◽

Vol 19 (1) ◽

Cited By ~ 7

Author(s):

Walter Carballosa ◽

Domingo Pestana ◽

José M. Rodríguez ◽

José M. Sigarreta

Keyword(s):

Metric Space ◽

Quantitative Information ◽

The Other ◽

Geodesic Triangle ◽

Interesting Topic ◽

Geodesic Metric Space ◽

Geodesic Metric ◽

Hyperbolic Graphs ◽

Two Sides ◽

Hyperbolicity Constant

If $X$ is a geodesic metric space and $x_1,x_2,x_3\in X$, a geodesic triangle $T=\{x_1,x_2,x_3\}$ is the union of the three geodesics $[x_1x_2]$, $[x_2x_3]$ and $[x_3x_1]$ in $X$. The space $X$ is $\delta$-hyperbolic $($in the Gromov sense$)$ if any side of $T$ is contained in a $\delta$-neighborhood of the union of the other two sides, for every geodesic triangle $T$ in $X$. We denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, i.e., $\delta(X):=\inf\{\delta\ge 0: \, X \, \text{ is $\delta$-hyperbolic}\,\}$. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. One of the main aims of this paper is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph $G\setminus e$ obtained from the graph $G$ by deleting an arbitrary edge $e$ from it. These inequalities allow to obtain the other main result of this paper, which characterizes in a quantitative way the hyperbolicity of any graph in terms of local hyperbolicity.

Download Full-text