Sequential coarse structures of topological groups

I. V. Protasov

doi:10.15330/ms.51.1.12-18

We endow a topological group $(G, \tau)$ with a coarse structure defined by the smallest group ideal $S_{\tau} $ on $G$ containing all converging sequences and denote the obtained coarse group by $(G, S_{\tau})$. If $G$ is discrete, then $(G, S_{\tau})$ is a finitary coarse group studding in {\it Geometric Group Theory}. The main result: if a topological abelian group $(G, \tau)$ contains a non-trivial converging sequence then {\it asdim} $(G, S_{\tau})= \infty $. We study metrizability, normality and functional boundedness of sequential coarse groups and put some open questions.

1. T. Banakh, I. Protasov, The normality and bounded growth of balleans, arXiv: 1810.07979.

2. T. Banakh, I. Protasov, Functional boundedness of balleans: coarse versions of compactness, Axioms, 8 (2019), №1, 33.

3. D. Dikranjan, I. Protasov, Coarse structures on groups defined by T-sequences, arXiv: 1902.02320.

4. D. Dikranjan, N. Zava, The impact of Pontryagin and Bohr functors on large-scale properties of LCAgroups, preprint.

5. P. de la Harpe, Topics in geometric group theory, Chicago Lectures in Math, University of Chicago Press, Chicago, 2000.

6. S. Hernandes, I.V. Protasov, Balleans of topological groups, J. Math. Sciences, 178 (2011), №1, 65–74.

7. A. Nikas, D. Rosenthal, On the asymptotic dimension of the dual group of a locally compact abelian group, Topology Appl., 160 (2013), 682–684.

8. I.V. Protasov, Normal ball structures, Mat. Stud., 20 (2003), №1, 3–16.

9. I. Protasov, T. Banakh, Ball Structures and Colorings of Groups and Graphs, Mat. Stud. Monogr. Ser 11, VNTL, Lviv, 2003.

10. I.V. Protasov, O.I. Protasova, Sketch of group balleans, Mat. Stud., 22 (2004), №1, 10–20.

11. I. Protasov, S. Slobodianiuk, On asymorphisms of groups, J. Group Theory, 20 (2017), 393–399.

12. I. Protasov, M. Zarichnyi, General asymptopogy, Math. Stud. Monogr. V.12, VNTL, Lviv, 2007.

13. I. Protasov, E. Zelenyuk, Topologies on Groups Determined by Sequences, Math. Stud. Monogr. Ser., V.4, VNTL, Lviv, 1999, 200 p.

14. J. Roe, Lectures on coarse geometry, Univ. Lecture Ser., V.31, American Mathematical Society, Providence RI, 2003.

	Sequential coarse structures of topological groups
Author	I. V. Protasov i.v.protasov@gmail.com Faculty of Computer Science and Cybernetics, Kyiv University Kyiv, Ukraine
Abstract	We endow a topological group $(G, \tau)$ with a coarse structure defined by the smallest group ideal $S_{\tau} $ on $G$ containing all converging sequences and denote the obtained coarse group by $(G, S_{\tau})$. If $G$ is discrete, then $(G, S_{\tau})$ is a finitary coarse group studding in {\it Geometric Group Theory}. The main result: if a topological abelian group $(G, \tau)$ contains a non-trivial converging sequence then {\it asdim} $(G, S_{\tau})= \infty $. We study metrizability, normality and functional boundedness of sequential coarse groups and put some open questions.
Keywords	coarse structure; group ideal; asymptotic dimension; Hamming space
DOI	doi:10.15330/ms.51.1.12-18
Reference	1. T. Banakh, I. Protasov, The normality and bounded growth of balleans, arXiv: 1810.07979. 2. T. Banakh, I. Protasov, Functional boundedness of balleans: coarse versions of compactness, Axioms, 8 (2019), №1, 33. 3. D. Dikranjan, I. Protasov, Coarse structures on groups defined by T-sequences, arXiv: 1902.02320. 4. D. Dikranjan, N. Zava, The impact of Pontryagin and Bohr functors on large-scale properties of LCAgroups, preprint. 5. P. de la Harpe, Topics in geometric group theory, Chicago Lectures in Math, University of Chicago Press, Chicago, 2000. 6. S. Hernandes, I.V. Protasov, Balleans of topological groups, J. Math. Sciences, 178 (2011), №1, 65–74. 7. A. Nikas, D. Rosenthal, On the asymptotic dimension of the dual group of a locally compact abelian group, Topology Appl., 160 (2013), 682–684. 8. I.V. Protasov, Normal ball structures, Mat. Stud., 20 (2003), №1, 3–16. 9. I. Protasov, T. Banakh, Ball Structures and Colorings of Groups and Graphs, Mat. Stud. Monogr. Ser 11, VNTL, Lviv, 2003. 10. I.V. Protasov, O.I. Protasova, Sketch of group balleans, Mat. Stud., 22 (2004), №1, 10–20. 11. I. Protasov, S. Slobodianiuk, On asymorphisms of groups, J. Group Theory, 20 (2017), 393–399. 12. I. Protasov, M. Zarichnyi, General asymptopogy, Math. Stud. Monogr. V.12, VNTL, Lviv, 2007. 13. I. Protasov, E. Zelenyuk, Topologies on Groups Determined by Sequences, Math. Stud. Monogr. Ser., V.4, VNTL, Lviv, 1999, 200 p. 14. J. Roe, Lectures on coarse geometry, Univ. Lecture Ser., V.31, American Mathematical Society, Providence RI, 2003.
Pages	12-18
Volume	51
Issue	1
Year	2019
Journal	Matematychni Studii
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