Balleans of bounded geometry and G-spaces

I.V.Protasov

A ballean (or a coarse structure) is a set endowed with some family of subsets which are called the balls. The properties of the family of balls are postulated in such a way that a ballean can be considered as an asymptotical counterpart of a uniform topological space. We prove that every ballean of bounded geometry is coarsely equivalent to a ballean on some set $X$ determined by some group of permutations of $X$.

1. A. Dranishnikov, Asymptotic topology, Russian Math. Surveys, 55(2000), 1085-1129.

2. M. Gromov, Asymptotic invariants for infinite groups, in Geometric Group Theory, vol.2, 1-295, Cambridge University Press, 1993.

3. F. Harary, Graph Theory, Addison-Wesley Publ. Comp., 1969.

4. P. Harpe, Topics in Geometrical Group Theory, University Chicago Press, 2000.

5. V. Nekrashevych, Uniformly bounded spaces, Problems in Algebra, 14, 47-67, Gomel University Press, 1999.

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

7. I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., vol.12, VNTL, Lviv, 2007.

8. J. Roe, Lectures on Coarse Geometry, University Lecture Series, vol.31, American Mathematical Society, Providence, RI, 2003.

	Balleans of bounded geometry and G-spaces
Author	I.V.Protasov protasov@unicyb.kiev.ua Department of Cybernetics, Kyiv Taras Shevchenko University,Volodimirska str., 64, 01033 Kyiv, Ukraine
Abstract	A ballean (or a coarse structure) is a set endowed with some family of subsets which are called the balls. The properties of the family of balls are postulated in such a way that a ballean can be considered as an asymptotical counterpart of a uniform topological space. We prove that every ballean of bounded geometry is coarsely equivalent to a ballean on some set $X$ determined by some group of permutations of $X$.
Keywords	ballean, bounded geometry, G-space
DOI	doi:10.30970/ms.30.1.61-66
Reference	1. A. Dranishnikov, Asymptotic topology, Russian Math. Surveys, 55(2000), 1085-1129. 2. M. Gromov, Asymptotic invariants for infinite groups, in Geometric Group Theory, vol.2, 1-295, Cambridge University Press, 1993. 3. F. Harary, Graph Theory, Addison-Wesley Publ. Comp., 1969. 4. P. Harpe, Topics in Geometrical Group Theory, University Chicago Press, 2000. 5. V. Nekrashevych, Uniformly bounded spaces, Problems in Algebra, 14, 47-67, Gomel University Press, 1999. 6. I. Protasov, T. Banakh, Ball Structures and Colorings of Groups and Graphs, Math. Stud. Monogr. Ser., vol.11, VNTL, Lviv, 2003. 7. I. Protasov, M. Zarichnyi, General Asymptology, Math. Stud. Monogr. Ser., vol.12, VNTL, Lviv, 2007. 8. J. Roe, Lectures on Coarse Geometry, University Lecture Series, vol.31, American Mathematical Society, Providence, RI, 2003.
Pages	61-66
Volume	30
Issue	1
Year	2008
Journal	Matematychni Studii
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