scholarly journals Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups

2015 ◽  
Vol 58 (4) ◽  
pp. 787-798 ◽  
Author(s):  
Yu Kitabeppu ◽  
Sajjad Lakzian
Keyword(s):  
Diameter Growth ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Finite Generation ◽  
Alexandrov Spaces ◽  
Geodesic Spaces ◽  
Special Case ◽  
Linear Diameter

AbstractIn this paper, we generalize the finite generation result of Sormani to non-branching RCD(0, N) geodesic spaces (and in particular, Alexandrov spaces) with full supportmeasures. This is a special case of the Milnor’s Conjecture for complete non-compact RCD(0, N) spaces. One of the key tools we use is the Abresch–Gromoll type excess estimates for non-smooth spaces obtained by Gigli–Mosconi.

scholarly journals Hyperbolic Groups are Hyperhopfian

2000 ◽  
Vol 68 (1) ◽  
pp. 126-130 ◽  
Author(s):  
Robert J. Daverman
Keyword(s):  
Abelian Subgroup ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Torsion Free

AbstractThe main result indicates that every finitely generated, residually finite, torsion-free, cohopfian group having on free Abelian subgroup of rank two is hyperhopfian. The argument relies on earlier work and ideas of Hirshon. As a corollary, fundamental groups of all closed hyperbolic manifolds are hyperhopfian.

scholarly journals Generators and relations of Rees matrix semigroups

1999 ◽  
Vol 42 (3) ◽  
pp. 481-495 ◽  
Author(s):  
H. Ayik ◽  
N. Ruškuc
Keyword(s):  
Rees Matrix Semigroup ◽  
Finitely Generated ◽  
Finite Generation ◽  
Generators And Relations ◽  
Rees Matrix Semigroups ◽  
Finite Presentability ◽  
Matrix Semigroup ◽  
Finitely Presented ◽  
Matrix Semigroups ◽  
The Ideal

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P.

Finite presentability of generalized Bruck–Reilly ∗-extension of groups

2016 ◽  
Vol 09 (04) ◽  
pp. 1650090 ◽  
Author(s):  
Seda Oğuz ◽  
Eylem G. Karpuz
Keyword(s):  
Finite Subset ◽  
Identity Element ◽  
Finitely Generated ◽  
Finite Generation ◽  
Finite Set ◽  
Finite Presentability ◽  
Finitely Presented

In [Finite presentability of Bruck–Reilly extensions of groups, J. Algebra 242 (2001) 20–30], Araujo and Ruškuc studied finite generation and finite presentability of Bruck–Reilly extension of a group. In this paper, we aim to generalize some results given in that paper to generalized Bruck–Reilly ∗-extension of a group. In this way, we determine necessary and sufficent conditions for generalized Bruck–Reilly ∗-extension of a group, [Formula: see text], to be finitely generated and finitely presented. Let [Formula: see text] be a group, [Formula: see text] be morphisms and [Formula: see text] ([Formula: see text] and [Formula: see text] are the [Formula: see text]- and [Formula: see text]-classes, respectively, contains the identity element [Formula: see text] of [Formula: see text]). We prove that [Formula: see text] is finitely generated if and only if there exists a finite subset [Formula: see text] such that [Formula: see text] is generated by [Formula: see text]. We also prove that [Formula: see text] is finitely presented if and only if [Formula: see text] is presented by [Formula: see text], where [Formula: see text] is a finite set and [Formula: see text] [Formula: see text] for some finite set of relations [Formula: see text].

scholarly journals Presentability by products for some classes of groups

2017 ◽  
Vol 09 (01) ◽  
pp. 27-49
Author(s):  
P. de la Harpe ◽  
D. Kotschick
Keyword(s):  
Coxeter Groups ◽  
Cohomological Dimension ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Finitely Generated Groups ◽  
Dense Subgroups ◽  
Finite Index Subgroups ◽  
Virtual Cohomological Dimension ◽  
By Products

In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of small virtual cohomological dimension and irreducible Zariski dense subgroups of appropriate algebraic groups. This leads to applications to groups of positive deficiency, to fundamental groups of three-manifolds and to Coxeter groups. For finitely generated groups presentable by products we discuss the problem of whether the factors in a presentation by products may be chosen to be finitely generated.

Brauer Characters and Grothendieck Rings

1975 ◽  
Vol 27 (5) ◽  
pp. 1025-1028 ◽  
Author(s):  
B. M. Puttaswamaiah
Keyword(s):  
Finite Order ◽  
Group Algebra ◽  
Irreducible Representations ◽  
Splitting Field ◽  
Finitely Generated ◽  
Grothendieck Ring ◽  
Primitive Idempotents ◽  
Brauer Characters ◽  
Grothendieck Rings ◽  
Special Case

Let G be a group of finite order g, A a splitting field of G of characteristic p (which may be 0) and R = AG the group algebra of G over A. In [2], the author studied some of the properties of the Grothendieck ring K(R) of the category of all finitely generated R-modules, and derived a number of consequences. This paper continues the study carried out in [2]. The study is concerned with the structure and irreducible representations of K(R). The ring K(R) is proved to be semisimple and the primitive idempotents of K(R) are explicitly constructed. If the ring K(R) is identified with the ‘algebra of representations', then Robinson's idempotent [3; 4; 5] follow from our description as a special case.

Finite generation of the cohomology rings for the extension algebras of monomial algebras

2022 ◽  
Author(s):  
Hongbo Shi
Keyword(s):  
Cohomology Ring ◽  
Minimal Resolution ◽  
Finitely Generated ◽  
Finite Generation ◽  
Cohomology Rings ◽  
Monomial Algebra ◽  
Monomial Algebras ◽  
Resolution Graph ◽  
Dual Extension

We describe the cohomology ring of a monomial algebra in the language of dimension tree or minimal resolution graph and in this context we study the finite generation of the cohomology rings of the extension algebras, showing among others that the cohomology ring [Formula: see text] is finitely generated [Formula: see text] is [Formula: see text] is, where [Formula: see text] is the dual extension of a monomial algebra [Formula: see text] and [Formula: see text] is the opposite algebra of [Formula: see text].

On the finite generation of high dimensional cohomology ring of virtually torsion-free groups

1998 ◽  
Vol 124 (1) ◽  
pp. 57-72 ◽  
Author(s):  
CHUN-NIP LEE
Keyword(s):  
Automorphism Groups ◽  
Cohomology Ring ◽  
Free Groups ◽  
Outer Automorphism ◽  
Cohomological Dimension ◽  
Group Cohomology ◽  
Mapping Class ◽  
Finitely Generated ◽  
Finite Generation ◽  
A Finite Group

Let Γ be a discrete group and p be a prime. One of the fundamental results in group cohomology is that H*(Γ, [ ]p) is a finitely generated [ ]p-algebra if Γ is a finite group [8, 24]. The purpose of this paper is to study the analogous question when Γ is no longer finite.Recall that Γ is said to have finite virtual cohomological dimension (vcd) if there exists a finite index torion-free subgroup Γ′ of Γ such that Γ′ has finite cohomological dimension over ℤ [4]. By definition vcd Γ is the cohomological dimension of Γ′. It is easy to see that the mod p cohomology ring of a finite vcd-group does not have to be a finitely generated [ ]p-algebra in general. For instance, if Γ is a countably infinite free product of ℤ's, then H1(Γ, [ ]p) is not finite dimensional over [ ]p. The three most important classes of examples of finite vcd-groups in which the mod p cohomology ring is a finitely generated [ ]p-algebra are arithmetic groups [2], mapping class groups [9, 10] and outer automorphism groups of free groups [5]. In each of these examples, the proof of finite generation involves the construction of a specific Γ-complex with appropriate finiteness conditions. These constructions should be regarded as utilizing the geometry underlying these special classes of groups. In contrast, the result we prove will depend only on the algebraic structure of the group Γ.

scholarly journals Residually finite rationally p groups

2019 ◽  
Vol 22 (03) ◽  
pp. 1950016
Author(s):  
Thomas Koberda ◽  
Alexander I. Suciu
Keyword(s):  
Algebraic Geometry ◽  
Group Theory ◽  
Large Class ◽  
Algebraic Curves ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Residual Property ◽  
Residually Finite ◽  
Torsion Freeness ◽  
Do So

In this paper, we develop the theory of residually finite rationally [Formula: see text] (RFR[Formula: see text]) groups, where [Formula: see text] is a prime. We first prove a series of results about the structure of finitely generated RFR[Formula: see text] groups (either for a single prime [Formula: see text], or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFR[Formula: see text] groups, which we use to study the boundary manifolds of algebraic curves [Formula: see text] and in [Formula: see text]. We show that boundary manifolds of a large class of curves in [Formula: see text] (which includes all line arrangements) have RFR[Formula: see text] fundamental groups, whereas boundary manifolds of curves in [Formula: see text] may fail to do so.

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