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 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.

The geometric dimension of an equivalence relation and finite extensions of countable groups

2009 ◽  
Vol 29 (6) ◽  
pp. 1789-1814
Author(s):  
A. H. DOOLEY ◽  
V. YA. GOLODETS
Keyword(s):  
Equivalence Relation ◽  
Free Groups ◽  
Geometric Dimension ◽  
Finite Extension ◽  
Countable Group ◽  
Finitely Generated ◽  
Free Products ◽  
Finitely Generated Groups ◽  
Equivalence Theory ◽  
Borel Probability

AbstractWe say that the geometric dimension of a countable group G is equal to n if any free Borel action of G on a standard Borel probability space (X,μ), induces an equivalence relation of geometric dimension n on (X,μ) in the sense of Gaboriau. Let ℬ be the set of all finitely generated amenable groups all of whose subgroups are also finitely generated, and let 𝒜 be the subset of ℬ consisting of finite groups, torsion-free groups and their finite extensions. In this paper we study finite free products K of groups in 𝒜. The geometric dimension of any such group K is one: we prove that also geom-dim(Gf(K))=1 for any finite extension Gf(K) of K, applying the results of Stallings on finite extensions of free product groups, together with the results of Gaboriau and others in orbit equivalence theory. Using results of Karrass, Pietrowski and Solitar we extend these results to finite extensions of free groups. We also give generalizations and applications of these results to groups with geometric dimension greater than one. We construct a family of finitely generated groups {Kn}n∈ℕ,n>1, such that geom-dim(Kn)=n and geom-dim(Gf(Kn))=n for any finite extension Gf(Kn) of Kn. In particular, this construction allows us to produce, for each integer n>1, a family of groups {K(s,n)}s∈ℕ of geometric dimension n, such that any finite extension of K(s,n) also has geometric dimension n, but the finite extensions Gf(K(s,n)) are non-isomorphic, if s≠s′.

On the Structure of Q2(G) for Finitely Generated Groups

1973 ◽  
Vol 25 (2) ◽  
pp. 353-359 ◽  
Author(s):  
Gerald Losey
Keyword(s):  
Abelian Group ◽  
Group Ring ◽  
Lower Central Series ◽  
Integral Group Ring ◽  
Symmetric Power ◽  
Central Series ◽  
Augmentation Ideal ◽  
Integral Group ◽  
Finitely Generated ◽  
Finitely Generated Groups

Let G be a group, ZG its integral group ring and Δ = Δ(G) the augmentation ideal of ZG. Denote by Gi the ith term of the lower central series of G. Following Passi [3], we set . It is well-known that (see, for example [1]). In [3] Passi shows that if G is an abelian group then , the second symmetric power of G.

scholarly journals The Herzog–Schönheim conjecture for finitely generated groups

2019 ◽  
Vol 29 (06) ◽  
pp. 1083-1112 ◽  
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.

scholarly journals Subshifts and colorings on ascending HNN-extensions of finitely generated abelian groups

2021 ◽  
pp. 1-26
Author(s):  
EDUARDO SILVA
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Global Constraints ◽  
Finitely Generated ◽  
Subshift Of Finite Type ◽  
Mixing Properties ◽  
Hnn Extensions ◽  
Hnn Extension ◽  
Periodic Configuration ◽  
The Right

Abstract For an ascending HNN-extension $G*_{\psi }$ of a finitely generated abelian group G, we study how a synchronization between the geometry of the group and weak periodicity of a configuration in $\mathcal {A}^{G*_{\psi }}$ forces global constraints on it, as well as in subshifts containing it. A particular case are Baumslag–Solitar groups $\mathrm {BS}(1,N)$ , $N\ge 2$ , for which our results imply that a $\mathrm {BS}(1,N)$ -subshift of finite type which contains a configuration with period $a^{N^\ell }\!, \ell \ge 0$ , must contain a strongly periodic configuration with monochromatic $\mathbb {Z}$ -sections. Then we study proper n-colorings, $n\ge 3$ , of the (right) Cayley graph of $\mathrm {BS}(1,N)$ , estimating the entropy of the associated subshift together with its mixing properties. We prove that $\mathrm {BS}(1,N)$ admits a frozen n-coloring if and only if $n=3$ . We finally suggest generalizations of the latter results to n-colorings of ascending HNN-extensions of finitely generated abelian groups.

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 A Tits alternative for topological full groups

2019 ◽  
Vol 41 (2) ◽  
pp. 622-640
Author(s):  
NÓRA GABRIELLA SZŐKE
Keyword(s):  
Free Group ◽  
Free Action ◽  
The Other ◽  
Full Group ◽  
Finitely Generated ◽  
Cantor Space ◽  
Cyclic Groups ◽  
Finitely Generated Groups ◽  
Tits Alternative ◽  
The One

We prove a Tits alternative for topological full groups of minimal actions of finitely generated groups. On the one hand, we show that topological full groups of minimal actions of virtually cyclic groups are amenable. By doing so, we generalize the result of Juschenko and Monod for $\mathbf{Z}$-actions. On the other hand, when a finitely generated group $G$ is not virtually cyclic, then we construct a minimal free action of $G$ on a Cantor space such that the topological full group contains a non-abelian free group.

scholarly journals Transparencia (judicial) = Judicial Transparency

2019 ◽  
pp. 198 ◽  
Author(s):  
Alejandro Coteño Muñoz
Keyword(s):  
Systematic Study ◽  
The Other ◽  
General Council ◽  
Access To Information ◽  
Future Studies ◽  
Other Hand ◽  
The One ◽  
The Right ◽  
Right Of Access

Resumen: La transparencia judicial representa una de las más importantes demandas de la ciudadanía a la Justicia. Tanto desde las instituciones como desde la Jurisprudencia y la Doctrina se ha afirmado repetidas veces la importancia de esta transparencia, sin embargo, hasta el momento, no se ha realizado un estudio profundo y sistemático de la misma. Es por ello que este texto trata de ahondar en el concepto de “transparencia judicial” y fijar los términos para futuros estudios y reformas. Así, por una parte, se analiza la publicidad activa −la publicación de información− y, por otra parte, la publicidad pasiva −el derecho de acceso a la información−, todo ello diferenciando si se trata de información relativa al Consejo General del Poder Judicial (en adelante, CGPJ) o de información judicial en sentido estricto. Para concluir, se aportan unas conclusiones que defienden la necesidad de reformas que ensanchen la transparencia judicial a fin de no dejarla en simples palabras.Palabras clave: Publicidad activa, publicidad pasiva, derecho de acceso, interesado, rendición de cuentas.Abstract: Judicial transparency represents one of the most important demands from the citizenship to Justice. From the institutions, as well as from Jurisprudence and Doctrine, the importance of this transparency has been repeatedly declared, however, until now, a deep and systematic study of it has not been carried out. That is why this text tries to delve into the concept of “judicial transparency” and set the terms for future studies and reforms. Thus, on the one hand, active transparency is analyzed −the publishing of information− and, on the other hand, passive transparency −the right of access to information−, all this distinguishing between information related to the General Council of the Judiciary or judicial information strictly talking. To end up, conclusions, which defend the need of reforms that broaden judicial transparency, so as not to leave it in simple words, are provided.Keywords: Active transparency, passive transparency, right of access, interested, accountability.

scholarly journals ON GROUPS AND COUNTER AUTOMATA

2008 ◽  
Vol 18 (08) ◽  
pp. 1345-1364 ◽  
Author(s):  
MURRAY ELDER ◽  
MARK KAMBITES ◽  
GRETCHEN OSTHEIMER
Keyword(s):  
Abelian Group ◽  
Word Problem ◽  
Word Problems ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Precise Sense

We study finitely generated groups whose word problems are accepted by counter automata. We show that a group has word problem accepted by a blind n-counter automaton in the sense of Greibach if and only if it is virtually free abelian of rank n; this result, which answers a question of Gilman, is in a very precise sense an abelian analogue of the Muller–Schupp theorem. More generally, if G is a virtually abelian group then every group with word problem recognized by a G-automaton is virtually abelian with growth class bounded above by the growth class of G. We consider also other types of counter automata.

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