scholarly journals Solvable Subgroup Theorem for simplicial nonpositive curvature

2018 ◽  
Vol 28 (04) ◽  
pp. 605-611
Author(s):  
Tomasz Prytuła
Keyword(s):  
Nonpositive Curvature ◽  
Decomposition Theorem ◽  
Solvable Subgroup ◽  
Finitely Generated ◽  
Flat Torus ◽  
Product Decomposition ◽  
Subgroup Theorem

Given a group [Formula: see text] with bounded torsion that acts properly on a systolic complex, we show that every solvable subgroup of [Formula: see text] is finitely generated and virtually abelian of rank at most [Formula: see text]. In particular, this gives a new proof of the above theorem for systolic groups. The main tools used in the proof are the Product Decomposition Theorem and the Flat Torus Theorem.

FREE INVERSE MONOIDS AND GRAPH IMMERSIONS

1993 ◽  
Vol 03 (01) ◽  
pp. 79-99 ◽  
Author(s):  
STUART W. MARGOLIS ◽  
JOHN C. MEAKIN
Keyword(s):  
Formal Language ◽  
Inverse Monoid ◽  
Finitely Generated ◽  
Inverse Monoids ◽  
Rational Subset ◽  
The Relationship ◽  
Subgroup Theorem ◽  
Free Inverse Monoid ◽  
Natural Way

The relationship between covering spaces of graphs and subgroups of the free group leads to a rapid proof of the Nielsen-Schreier subgroup theorem. We show here that a similar relationship holds between immersions of graphs and closed inverse submonoids of free inverse monoids. We prove using these methods, that a closed inverse submonoid of a free inverse monoid is finitely generated if and only if it has finite index if and only if it is a rational subset of the free inverse monoid in the sense of formal language theory. We solve the word problem for the free inverse category over a graph Γ. We show that immersions over Γ may be classified via conjugacy classes of loop monoids of the free inverse category over Γ. In the case that Γ is a bouquet of X circles, we prove that the category of (connected) immersions over Γ is equivalent to the category of (transitive) representations of the free inverse monoid FIM(X). Such representations are coded by closed inverse submonoids of FIM(X). These monoids will be constructed in a natural way from groups acting freely on trees and they admit an idempotent pure retract onto a free inverse monoid. Applications to the classification of finitely generated subgroups of free groups via finite inverse monoids are developed.

Direct product decomposition of theories of modules

1979 ◽  
Vol 44 (1) ◽  
pp. 77-88 ◽  
Author(s):  
Steven Garavaglia
Keyword(s):  
Direct Product ◽  
Regular Ring ◽  
Decomposition Theorem ◽  
Direct Products ◽  
Product Decomposition ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Direct Product Decomposition ◽  
Von Neumann Regular ◽  
Complete Theories

This paper is mainly concerned with describing complete theories of modules by decomposing them (up to elementary equivalence) into direct products of simpler modules. In §1, I give a decomposition theorem which works for arbitrary direct product theories T. Given such a T, I define T-indecomposable structures and show that every model of T is elementarily equivalent to a direct product of T-indecomposable models of T. In §2, I show that if R is a commutative ring then every R-module is elementarily equivalent to ΠMM where M ranges over the maximal ideals of R and M is the localization of at M. This is applied to prove that if R is a commutative von Neumann regular ring and TR is the theory of R-modules then the TR-indecomposables are precisely the cyclic modules of the form R/M where M is a maximal ideal. In §3, I use the decomposition established in §2 to characterize the ω1-categorical and ω-stable modules over a countable commutative von Neumann regular ring and the superstable modules over a commutative von Neumann regular ring of arbitrary cardinality. In the process, I also prove several general characterizations of ω-stable and superstable modules; e.g., if R is any countable ring, then an R-moduIe is ω-stable if and only if every R-module elementarily equivalent to it is equationally compact.

scholarly journals Flats in Spaces with Convex Geodesic Bicombings

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Dominic Descombes ◽  
Urs Lang
Keyword(s):  
Distance Function ◽  
Metric Spaces ◽  
Nonpositive Curvature ◽  
Flat Torus ◽  
Convexity Assumptions ◽  
Weaker Conditions

AbstractIn spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales.We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function along distinguished geodesics. The discussion includes, among other things, the Flat Torus Theorem and Gromov’s hyperbolicity criterion referring to embedded planes. This generalizes results of Bowditch for Busemann spaces.

A Decomposition of Rings Generated by Faithful Cyclic Modules

1989 ◽  
Vol 32 (3) ◽  
pp. 333-339 ◽  
Author(s):  
Gary F. Birkenmeier
Keyword(s):  
Decomposition Theorem ◽  
Commutative Rings ◽  
Finitely Generated ◽  
Nilpotent Elements ◽  
Strongly Regular ◽  
Regular Rings

AbstractA ring R is said to be generated by faithful right cyclics (right finitely pseudo-Frobenius), denoted by GFC (FPF), if every faithful cyclic (finitely generated) right R-module generates the category of right R-modules. The class of right GFC rings includes right FPF rings, commutative rings (thus every ring has a GFC subring - its center), strongly regular rings, and continuous regular rings of bounded index. Our main results are: (1) a decomposition of a semi-prime quasi-Baer right GFC ring (e.g., a semiprime right FPF ring) is achieved by considering the set of nilpotent elements and the centrality of idempotnents; (2) a generalization of S. Page's decomposition theorem for a right FPF ring.

Quantum isometry groups of dual of finitely generated discrete groups and quantum groups

2017 ◽  
Vol 29 (03) ◽  
pp. 1750008 ◽  
Author(s):  
Debashish Goswami ◽  
Arnab Mandal
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Polynomial Growth ◽  
Finitely Generated ◽  
Isometry Groups ◽  
Product Decomposition ◽  
Full Spectrum ◽  
Spectral Triples ◽  
Growth Property ◽  
Quantum Isometry Groups

We study quantum isometry groups, denoted by [Formula: see text], of spectral triples on [Formula: see text] for a finitely generated discrete group [Formula: see text] coming from the word-length metric with respect to a symmetric generating set [Formula: see text]. We first prove a few general results about [Formula: see text] including: • For a group [Formula: see text] with polynomial growth property, the dual of [Formula: see text] has polynomial growth property provided the action of [Formula: see text] on [Formula: see text] has full spectrum. •[Formula: see text] for any discrete abelian group [Formula: see text], where [Formula: see text] is a suitable metric on the dual compact abelian group [Formula: see text]. We then carry out explicit computations of [Formula: see text] for several classes of examples including free and direct product of cyclic groups, Baumslag–Solitar group, Coxeter groups etc. In particular, we have computed quantum isometry groups of all finitely generated abelian groups which do not have factors of the form [Formula: see text] or [Formula: see text] for some [Formula: see text] in the direct product decomposition into cyclic subgroups.

scholarly journals RIGIDITY OF GRAPH PRODUCTS OF ABELIAN GROUPS

2008 ◽  
Vol 77 (2) ◽  
pp. 187-196 ◽  
Author(s):  
MAURICIO GUTIERREZ ◽  
ADAM PIGGOTT
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Abelian Groups ◽  
Graph Products ◽  
Vertex Group ◽  
Graph Product ◽  
Finitely Generated ◽  
Product Decomposition ◽  
Unique Decomposition ◽  
Product Of Groups

AbstractWe show that if G is a group and G has a graph-product decomposition with finitely generated abelian vertex groups, then G has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely generated abelian group and the graph satisfies the T0 property. Our results build on results by Droms, Laurence and Radcliffe.

Decomposition Theory of Perfect Modules

2009 ◽  
Vol 16 (01) ◽  
pp. 49-64
Author(s):  
Zhixiang Wu ◽  
K. P. Shum
Keyword(s):  
Decomposition Theorem ◽  
Primary Decomposition ◽  
Finitely Generated ◽  
Arbitrary Ring ◽  
Decomposition Theory ◽  
Torsion Modules ◽  
Special Case

The decomposition theory of perfect modules is developed and consequently the decomposition of comodules is extended to perfect modules over an arbitrary ring. In addition, a decomposition theorem of finitely generated torsion modules over PID is established and the theory of Dickson on the primary decomposition of modules becomes a special case of our result.

scholarly journals o-MINIMAL COHOMOLOGY: FINITENESS AND INVARIANCE RESULTS

2009 ◽  
Vol 09 (02) ◽  
pp. 167-182 ◽  
Author(s):  
ALESSANDRO BERARDUCCI ◽  
ANTONGIULIO FORNASIERO
Keyword(s):  
Betti Numbers ◽  
Decomposition Theorem ◽  
Cell Decomposition ◽  
Compact Set ◽  
Cohomology Groups ◽  
Finitely Generated ◽  
Minimal Cell ◽  
General Validity ◽  
Cw Complex ◽  
Definable Sets

The topology of definable sets in an o-minimal expansion of a group is not fully understood due to the lack of a triangulation theorem. Despite the general validity of the cell decomposition theorem, we do not know whether any definably compact set is a definable CW-complex. Moreover the closure of an o-minimal cell can have arbitrarily high Betti numbers. Nevertheless we prove that the cohomology groups of a definably compact set over an o-minimal expansion of a group are finitely generated and invariant under elementary extensions and expansions of the language.

Incompressible Surfaces in the Boundary of a Handlebody–an Algorithm

1980 ◽  
Vol 32 (3) ◽  
pp. 590-595 ◽  
Author(s):  
Herbert C. Lyon
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Decomposition Theorem ◽  
Finitely Generated ◽  
Initial Development ◽  
Generating Set ◽  
3 Dimensional ◽  
Incompressible Surfaces ◽  
Minimal Generating Set

Our first result is a decomposition theorem for free groups relative to a set of elements. This enables us to formulate several algebraic conditions, some necessary and some sufficient, for various surfaces in the boundary of a 3-dimensional handlebody to be incompressible. Moreover, we show that there exists an algorithm to determine whether or not these algebraic conditions are met.Many of our algebraic ideas are similar to those of Shenitzer [3]. Conversations with Professor Roger Lyndon were helpful in the initial development of these results, and he reviewed an earlier version of this paper, suggesting Theorem 1 (iii) and its proof. Our notation and techniques are standard (cf. [1], [2]). A set X of elements in a finitely generated free group F is a basis if it is a minimal generating set, and X±l denotes the set of all elements in X, together wTith their inverses.

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