Completion of the group topology of a countable group

1990 ◽  
Vol 31 (1) ◽  
pp. 1-10 ◽  
Author(s):  
V. I. Arnautov ◽  
E. I. Kabanova
Keyword(s):  
Countable Group ◽  
Group Topology

scholarly journals L2-BETTI NUMBERS OF DISCRETE MEASURED GROUPOIDS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1169-1188 ◽  
Author(s):  
ROMAN SAUER
Keyword(s):  
Probability Space ◽  
Betti Numbers ◽  
Free Action ◽  
Countable Group ◽  
Equivalence Relations ◽  
Dimension Theory ◽  
Discrete Groups ◽  
Type Ii ◽  
Orbit Equivalent ◽  
Definition Of

There are notions of L2-Betti numbers for discrete groups (Cheeger–Gromov, Lück), for type II1-factors (recent work of Connes-Shlyakhtenko) and for countable standard equivalence relations (Gaboriau). Whereas the first two are algebraically defined using Lück's dimension theory, Gaboriau's definition of the latter is inspired by the work of Cheeger and Gromov. In this work we give a definition of L2-Betti numbers of discrete measured groupoids that is based on Lück's dimension theory, thereby encompassing the cases of groups, equivalence relations and holonomy groupoids with an invariant measure for a complete transversal. We show that with our definition, like with Gaboriau's, the L2-Betti numbers [Formula: see text] of a countable group G coincide with the L2-Betti numbers [Formula: see text] of the orbit equivalence relation [Formula: see text] of a free action of G on a probability space. This yields a new proof of the fact the L2-Betti numbers of groups with orbit equivalent actions coincide.

scholarly journals On the embedding of a group in a join of given groups

1974 ◽  
Vol 17 (4) ◽  
pp. 434-495 ◽  
Author(s):  
P. Hall
Keyword(s):  
Countable Group ◽  
The Other ◽  
Other Hand ◽  
First Time ◽  
Fundamental Paper

1. In their fundamental paper of 1949, Higman, Neumann and Neumann proved for the first time that a countable group can always be embedded in some 2-generator group: [1], Theorem IV. Two kinds of improvement of this result have recently appeared. In [4], Theorem 2, Dark has shown that the embedding can always be made subnormally. On the other hand, in [2], Theorem 2.1, Levin has shown that the two generators can be given preassigned orders m > 1 and n > 2; and in [3], Miller and Schupp prove that the 2-generator group can also be made to satisfy several additional requirements, such as being complete and Hopfian.

scholarly journals Borel structurability on the 2-shift of a countable group

2016 ◽  
Vol 167 (1) ◽  
pp. 1-21 ◽  
Author(s):  
Brandon Seward ◽  
Robin D. Tucker-Drob
Keyword(s):  
Countable Group

EVERY COUNTABLE GROUP HAS THE WEAK ROHLIN PROPERTY

2006 ◽  
Vol 38 (06) ◽  
pp. 932-936 ◽  
Author(s):  
E. GLASNER ◽  
J.-P. THOUVENOT ◽  
B. WEISS
Keyword(s):  
Countable Group

scholarly journals A nonlinear maximal group topology

2012 ◽  
Vol 229 (4) ◽  
pp. 2415-2426
Author(s):  
Yevhen Zelenyuk
Keyword(s):  
Group Topology ◽  
Maximal Group

scholarly journals A note on the duality of locally compact groups

1968 ◽  
Vol 9 (2) ◽  
pp. 87-91 ◽  
Author(s):  
J. W. Baker
Keyword(s):  
Abelian Group ◽  
Compact Group ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Group Topology ◽  
Locally Compact ◽  
Natural Way

Let H be a group of characters on an (algebraic) abelian group G. In a natural way, we may regard G as a group of characters on H. In this way, we obtain a duality between the two groups G and H. One may pose several problems about this duality. Firstly, one may ask whether there exists a group topology on G for which H is precisely the set of continuous characters. This question has been answered in the affirmative in [1]. We shall say that such a topology is compatible with the duality between G and H. Next, one may ask whether there exists a locally compact group topology on G which is compatible with a given duality and, if so, whether there is more than one such topology. It is this second question (previously considered by other authors, to whom we shall refer below) which we shall consider here.

scholarly journals Group, basic assumptions and complexity science

2018 ◽  
Vol 52 (1) ◽  
pp. 3-22 ◽  
Author(s):  
Giulio de Felice ◽  
Giuseppe De Vita ◽  
Alessandro Bruni ◽  
Assunta Galimberti ◽  
Giulia Paoloni ◽  
...  
Keyword(s):  
Critical Points ◽  
Work Group ◽  
Complexity Science ◽  
Group Topology ◽  
Basic Assumptions ◽  
Psychoanalytic Literature ◽  
Group Mentality

This article represents the first complete systematization of the basic assumptions as theorized by Wilfred R. Bion and post-Bionian authors. The authors reviewed, compared and systematized all the Bionian developments concerning the basic assumptions taking the prevailing anxieties, group topology, leader peculiarities, interactions with the work-group mentality into account. The analysis evinced five main ba(s) and five subsets (i.e. their features resemble one of the five main basic assumptions). Briefly, in the first paragraph the authors summarize Bionian thought and its underlying logical criteria while in the second they reviewed all the new proposals for basic assumptions emerging from the psychoanalytic literature (i.e. Lawrence, Bain and Gould, 1996; Romano, 1997; Sandler, 2002; Sarno, 1999; Turquet, 1974; Hopper, 2009). In conclusion the authors focus on the main strengths and critical points of the systematization. In the last section ‘Promising developments’ they address the methodology of the study of basic assumptions, its main features and potential developments. The article rounds off with a clinical appendix.

Embedding Theorems for Countable Groups

1970 ◽  
Vol 22 (4) ◽  
pp. 827-835 ◽  
Author(s):  
James McCool
Keyword(s):  
Free Product ◽  
Word Problem ◽  
Countable Group ◽  
Factor Group ◽  
Decision Problems ◽  
Alternative Proof ◽  
Embedding Theorems ◽  
Cyclic Groups ◽  
Result Theorem ◽  
The Given

A group P is said to be a CEF-group if, for every countable group G, there is a factor group of P which contains a subgroup isomorphic to G. It was shown by Higman, Neumann, and Neumann [5] that the free group of rank two is a CEF-group. More recently, Levin [6] proved that if P is the free product of two cyclic groups, not both of order two, then P is a CEF-group. Later, Hall [3] gave an alternative proof of Levin's result.In this paper we give a new proof of Levin's result (Theorem 2). The proof given yields information about the factor group H of P in which a given countable group G is embedded; for example, if G is given by a recursive presentation (this concept is denned in [4]), then a recursive presentation is obtained for H, and certain decision problems (in particular, the word problem) are solvable for the recursive presentation obtained for H if and only if they are solvable for the given recursive presentation of G.

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