Free coarse groups

2019 ◽  
Vol 22 (4) ◽  
pp. 775-782
Author(s):  
Igor Protasov ◽  
Ksenia Protasova
Keyword(s):  
Topological Groups ◽  
Coarse Structure ◽  
Variety Of Groups ◽  
And Inversion ◽  
Coarse Space

AbstractA coarse group is a group endowed with a coarse structure so that the group multiplication and inversion are coarse mappings. Let {(X,\mathcal{E})} be a coarse space, and let {\mathfrak{M}} be a variety of groups different from the variety of singletons. We prove that there is a coarse group {F_{\mathfrak{M}}(X,\mathcal{E})\in\mathfrak{M}} such that {(X,\mathcal{E})} is a subspace of {F_{\mathfrak{M}}(X,\mathcal{E})}, X generates {F_{\mathfrak{M}}(X,\mathcal{E})} and every coarse mapping {(X,\mathcal{E})\to(G,\mathcal{E}^{\prime})}, where {G\in\mathfrak{M}}, {(G,\mathcal{E}^{\prime})} is a coarse group, can be extended to coarse homomorphism {F_{\mathfrak{M}}(X,\mathcal{E})\to(G,\mathcal{E}^{\prime})}. If {\mathfrak{M}} is the variety of all groups, the groups {F_{\mathfrak{M}}(X,\mathcal{E})} are asymptotic counterparts of Markov free topological groups over Tikhonov spaces.

scholarly journals Coarse cohomology with twisted coefficients

2020 ◽  
Vol 70 (6) ◽  
pp. 1413-1444
Author(s):  
Elisa Hartmann
Keyword(s):  
Sheaf Cohomology ◽  
Grothendieck Topology ◽  
Coarse Structure ◽  
Relative Cohomology ◽  
Number Of Ends ◽  
Constant Coefficients ◽  
Coarse Space ◽  
Grothendieck Topologies

AbstractTo a coarse structure we associate a Grothendieck topology which is determined by coarse covers. A coarse map between coarse spaces gives rise to a morphism of Grothendieck topologies. This way we define sheaves and sheaf cohomology on coarse spaces. We obtain that sheaf cohomology is a functor on the coarse category: if two coarse maps are close they induce the same map in cohomology. There is a coarse version of a Mayer-Vietoris sequence and for every inclusion of coarse spaces there is a coarse version of relative cohomology. Cohomology with constant coefficients can be computed using the number of ends of a coarse space.

scholarly journals Extremal balleans

2019 ◽  
Vol 20 (1) ◽  
pp. 297
Author(s):  
Igor Protasov
Keyword(s):  
Coarse Structure ◽  
Unbounded Subset ◽  
Coarse Space ◽  
Extremal Properties

<p>A ballean (or coarse space) is a set endowed with a coarse structure. A ballean X is called normal if any two asymptotically disjoint subsets of X are asymptotically separated.  We say that a ballean X is ultra-normal (extremely normal) if any two unbounded subsets of X are not asymptotically disjoint (every unbounded subset of X is large).   Every maximal ballean is extremely normal and every extremely normal ballean is ultranormal, but the converse statements do not hold.   A normal ballean is ultranormal if and only if the Higson′s corona of X is a singleton.   A discrete ballean X is ultranormal if and only if X is maximal.  We construct a series of concrete balleans with extremal properties.</p>

TOPOLOGICAL GROUPS: VIRTUE OF PRE-OPEN SETS

2021 ◽  
Vol 10 (1) ◽  
pp. 385-389
Author(s):  
P. Gnanachandra ◽  
A.M Kumar
Keyword(s):  
Topological Group ◽  
Topological Groups ◽  
And Inversion

In this paper, we introduce notions of $\mathit{p}$-topological group and $\mathit{p}$-irresolute topological group which are generalizations of the notion topological group. We discuss the properties of $\mathit{p}$-topological group with illustrated examples. Also, we prove that translation and inversion in $\mathit{p}$-topological group are $\mathit{p}$-homeomorphism.

scholarly journals Hausdorffness in varieties of topological groups

1998 ◽  
Vol 57 (1) ◽  
pp. 147-151
Author(s):  
Carolyn E. McPhail
Keyword(s):  
Proper Class ◽  
Topological Groups ◽  
Variety Of Groups ◽  
Quotient Groups

A variety of topological groups is a class of (not necessarily Hausdorff) topological groups closed under the operations of forming subgroups, quotient groups and arbitrary products. It is well known that the class of groups underlying the topological groups contained in any variety of topological groups is a variety of groups. Much work on topological groups is restricted to Hausdorff topological groups and so it is relevant to know if the class of groups underlying Hausdorff topological groups in is a variety of groups. It is shown that this is not always the case. Indeed it is proved that this is not the case for an important proper class of varieties of topological groups.

scholarly journals On generalized topological groups

Filomat ◽  
2013 ◽  
Vol 27 (4) ◽  
pp. 567-575 ◽  
Author(s):  
Murad Hussain ◽  
ud Khan ◽  
Cenap Özel
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Algebraic Structure ◽  
Topological Structure ◽  
Topological Groups ◽  
Generalized Topological Space ◽  
And Inversion

In this paper, we initiate the study of generalized topological groups. A generalized topological group has the algebraic structure of groups and the topological structure of a generalized topological space defined by A. Cs?sz?r [2] and they are joined together by the requirement that multiplication and inversion are G-continuous. Every topological group is a G-topological group whereas converse is not true in general. Quotients of generalized topological groups are defined and studied.

scholarly journals Coarse structures on groups defined by conjugations

2021 ◽  
Vol 32 (1) ◽  
pp. 65-75
Author(s):  
I. Protasov ◽  
◽  
K. Protasova ◽  
Keyword(s):  
Asymptotic Properties ◽  
Infinite Group ◽  
Locally Finite ◽  
Coarse Structure ◽  
Dedekind Group ◽  
Algebraic Properties ◽  
Coarse Structures ◽  
Coarse Space

For a group G, we denote by G↔ the coarse space on G endowed with the coarse structure with the base {{(x,y)∈G×G:y∈xF}:F∈[G]<ω}, xF={z−1xz:z∈F}. Our goal is to explore interplays between algebraic properties of G and asymptotic properties of G↔. In particular, we show that asdim G↔=0 if and only if G/ZG is locally finite, ZG is the center of G. For an infinite group G, the coarse space of subgroups of G is discrete if and only if G is a Dedekind group.

scholarly journals Selectors and orderings of coarse spaces

2021 ◽  
Vol 18 (1) ◽  
pp. 71-79
Author(s):  
Igor Protasov
Keyword(s):  
Linear Orders ◽  
Coarse Structure ◽  
Coarse Space

Given a coarse space $(X, \mathcal{E})$, we consider linear orders on $X$ compatible with the coarse structure $\mathcal E$ and explore interplays between these orders and macro-uniform selectors of $(X, \mathcal{E})$.

TOPOLOGICAL GROUPS: VIRTUE OF PRE-OPEN SETS

2021 ◽  
Vol 10 (1) ◽  
pp. 385-389
Author(s):  
P. Gnanachandra ◽  
A.M Kumar
Keyword(s):  
Topological Group ◽  
Topological Groups ◽  
And Inversion

In this paper, we introduce notions of $\mathit{p}$-topological group and $\mathit{p}$-irresolute topological group which are generalizations of the notion topological group. We discuss the properties of $\mathit{p}$-topological group with illustrated examples. Also, we prove that translation and inversion in $\mathit{p}$-topological group are $\mathit{p}$-homeomorphism.

scholarly journals Closeness and linkness in balleans

2020 ◽  
Vol 53 (1) ◽  
pp. 100-108
Author(s):  
I.V. Protasov ◽  
K. Protasova
Keyword(s):  
General Question ◽  
Coarse Structure ◽  
Coarse Space

A set $X$ endowed with a coarse structure is called ballean or coarse space. For a ballean $(X, \mathcal{E})$, we say that two subsets $A$, $B$ of $X$ are close (linked) if there exists an entourage $E\in \mathcal{E}$ such that $A\subseteq E [B]$, $B\subseteq E[A]$ (either $A, B$ are bounded or contain unbounded close subsets). We explore the following general question: which information about a ballean is contained and can be extracted from the relations of closeness and linkness.

The Fine Structure of the Sheath of Volvox Carteri f. Weismannia

1977 ◽  
Vol 35 ◽  
pp. 346-347
Author(s):  
Regina Birchem
Keyword(s):  
Developmental Stages ◽  
Ruthenium Red ◽  
Sulfated Polysaccharides ◽  
Volvox Carteri ◽  
High Voltage Electron Microscopy ◽  
Ultrastructural Studies ◽  
Voltage Electron ◽  
Sheath Formation ◽  
And Inversion ◽  
Sheath Material

Spheroids of the green colonial alga Volvox consist of biflagellate Chlamydomonad-like cells embedded in a transparent sheath. The sheath, important as a substance through which metabolic materials, light, and the sexual inducer must pass to and from the cells, has been shown to have an ordered structure (1,2). It is composed of both protein and carbohydrate (3); studies of V. rousseletii indicate an outside layer of sulfated polysaccharides (4).Ultrastructural studies of the sheath material in developmental stages of V. carteri f. weismannia were undertaken employing variations in the standard fixation procedure, ruthenium red, diaminobenzidine, and high voltage electron microscopy. Sheath formation begins after the completion of cell division and inversion of the daughter spheroids. Golgi, rough ER, and plasma membrane are actively involved in phases of sheath synthesis (Fig. 1). Six layers of ultrastructurally differentiated sheath material have been identified.

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