Two questions, the first: What is the smallest non negative integer that we do not know yet is the Tarski number of a group? The second question is the same as in the title: What is the latest progress on Tarski numbers?
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2$\begingroup$ For the second question, I think you may find the following papers interesting: M. Ershov, G. Golan, M. Sapir. The Tarski numbers of groups, Advances in Mathematics, Vol. 284, 2015, 21-53; A. Rejali, A. Yousofzadeh. Configuration of groups and paradoxical decompositions. Bull. Belg. Math. Soc. Simon Stevin 18 (2011), no. 1, 157-172; A. Yousofzadeh, A constructive way to compute the Tarski number of a group, J. Algebra. Appl, 17, no. 1 (2018), 1850139 $\endgroup$– Manuel NormanCommented Jul 29, 2020 at 12:54
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$\begingroup$ @CarloBeenakker That is, we know that there is a group with Tarski number 5. Could you please provide a reference? $\endgroup$– MSMalekanCommented Jul 29, 2020 at 13:03
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1 Answer
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Q1: The state of the art as reported in 2014, see arXiv:1406.2097, is that the only numbers which are known to be Tarski numbers of some groups are 4,5,6. Tarski numbers $<4$ are forbidden, which suggests that 7 is the answer to the question "What is the smallest non negative integer that we do not know yet is the Tarski number of a group?".
Q2: Recent progress, in addition to the references by Manuel Norman: In arXiv:1603.04212 it is shown that the Tarski number of Burnside groups is in the range $[6,14]$.
answered Jul 29, 2020 at 13:09
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$\begingroup$ broken link fixed, thanks. $\endgroup$ Commented Jul 29, 2020 at 20:02