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Configuration sets; a right place for ping-pong arguments | ||
| Journal of Discrete Mathematics and Its Applications | ||
| مقاله 4، دوره 9، شماره 3، آذر 2024، صفحه 191-202 اصل مقاله (309.34 K) | ||
| نوع مقاله: Full Length Article | ||
| شناسه دیجیتال (DOI): 10.22061/jdma.2024.11140.1079 | ||
| نویسنده | ||
| Meisam Soleimani Malekan* | ||
| Department of Mathematics, Faculty of Science, Shahid Rajaee Teacher Training University | ||
| تاریخ دریافت: 16 مرداد 1403، تاریخ بازنگری: 07 شهریور 1403، تاریخ پذیرش: 09 شهریور 1403 | ||
| چکیده | ||
| Giving a condition for the amenability of groups, Rosenblatt and Willis first introduced the concept of configuration. In this paper, we investigate the relationship between ping-pong lemma and configuration sets, and show that only one configuration set is enough to ensure that several elements in a group generates a free subgroup of that group. Using only one two-sided configuration sets, we give, in a sense, a generalization of this result to polycyclic or FC-groups. Finiteness and paradoxical decompositions of groups, are other properties which can be characterized with only one configuration set. | ||
| کلیدواژه‌ها | ||
| Configuration؛ FC-Group؛ Finitely Presented Group؛ Polycyclic Group | ||
| مراجع | ||
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[1] Can infiniteness of finitely generated groups be read by a paradoxical decomposition? https://mathoverflow.net/questions.298045/can-infiniteness-of-finitely-generated-groupsbe-read-by-a-paradoxical-decompo.
[2] M. Soleimani Malekan, A. Rejali, Configuration equivalence is not equivalent to isomorphism, Internat. J. Algebra Comput. 27 (2017) 1073–1085.
[3] A. Abdollahi, A. Rejali, G. A. Willis, Group properties characterised by configurations, Illinois J. Math. 48 (2004) 861–873. [4] M.R.Bridson, A. Hafliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, Springer Berlin Heidelberg, 2013.
[5] P. de la Harpe, Topics in Geometric Group Theory, Chicago Lectures in Mathematics, University of Chicago Press, 2000. [6] R. C. Lyndon, P. E. Schup, Combinatorial Group Theory, Classics in Mathematics, Springer Berlin Heidelberg, 2015.
[7] M. Soleimani Malekan, A. Rejali, Two-sided conguration equivalence and isomorphism, Journal of Algebra and Its Applications 19 (2020) 2050189.
[8] A. Rejali, M. Soleimani Malekan, Solubility of groups can be characterized by configuration, New York Journal of Mathematics 23 (2017)1427–1445.
[9] A. Rejali, A. Yousofzadeh, Group properties characterized by two-sided configurations, Algebra Colloquium 17 (2010) 583–594.
[10] J. M. Rosenblatt, G. A. Willis, Weak convergence is not strong convergence for amenable groups, Canad. Math. Bull. 44 (2001) 231–241.
[11] V. Runde, Lectures on Amenability, Number1774inLecture NotesinMathematics, Springer, 2002.
[12] M. V. Sapir, V. S. Guba, M. V. Volkov, Combinatorial Algebra: Syntax and Semantics, Springer Monographs in Mathematics, Springer International Publishing, 2014.
[13] J. Tits, Free subgroups in linear groups, Journal of Algebra 20 (1972) 250–270. | ||
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