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We show that R.J. Thompson’s groups F and T are bi-interpretable with the ring of the integers. From a result by A. Khélif, these groups are quasi-finitely axiomatizable and prime. So, the group T provides an example of a simple group which is quasi-finitely axiomatizable and prime. This answers questions posed by T. Altınel and A. Muranov in [2], and by A. Nies in [12].



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[1]Ahlbrandt, G. and Ziegler, M., Quasi-finitely axiomatizable totally categorical theories. Annals of Pure and Applied Logic, vol. 30 (1986), pp. 6382.
[2]Altınel, T. and Muranov, A., Interprétation de l’arithmétique dans certains groupes de permutations affines par morceaux d’un intervalle. Journal of the Institute of Mathematics of Jussieu, vol. 8 (2009), pp. 623652.
[3]Bardakov, V. and Tolstykh, V., Interpreting the arithmetic in Thompson’s group F. Journal of Pure and Applied Algebra, vol. 211 (2007), pp. 633637.
[4]Cannon, J.W., Floyd, W.J., and Parry, W.R., Introductory notes on Richard Thompson’s groups, L’Enseignement Mathématique (2), vol. 42 (1996), pp. 215256.
[5]Davis, M., Hilbert’s tenth problem is unsolvable. American Mathematical Monthly, vol. 80 (1973), pp. 233269.
[6]Hodges, W., Model theory, Encyclopedia of mathematics and its applications, Cambridge University Press, Cambridge, 1993.
[7]Khélif, A., Bi-interprétabilité et structures QFA : étude de groupes résolubles et des anneaux commutatifs, Comptes Rendus Mathématique de l’Académie des Sciences de Paris, vol. 345 (2007), pp. 5961.
[8]Lasserre, C., Sur les groupes de type fini : primalité, axiomatisabilité quasi finie et bi-interprétabilité avec l’arithmétique, Ph.D. thesis, Université Paris 7, September 2011.
[9]Matijasevich, Y.V., Enumerable sets are diophantine, Soviet Mathematics Doklady, vol. 11 (1970), pp. 354358.
[10]Morozov, A. and Nies, A., Finitely generated groups and first order logic. Journal of London Mathematical Society, vol. 71 (2005), pp. 545562.
[11]Nies, A., Separating classes of groups by first-order sentences. International Journal of Algebra and Computation, vol. 13 (2003), pp. 287302.
[12]Nies, A., Comparing quasi-finitely axiomatizable groups and prime models. Journal of Group Theory, vol. 10 (2007), pp. 347361.
[13]Nies, A., Describing groups. Bulletin of Symbolic Logic, vol. 13 (2007), pp. 305339.
[14]Noskov, G.A., On the elementary theory of a finitely generated almost solvable group, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 47 (1983), pp. 498517.
[15]Odifreddi, P., Classical recursion theory, vol. 1, North-Holland, Amsterdam, 1999.
[16]Ould Houcine, A., Homogeneity and prime models in torsion-free hyperbolic groups. Confluentes Mathematici, vol. 3 (2011), pp. 121155.
[17]Rhemtulla, A.H., Commutators of certain finitely generated soluble groups. Canadian Journal of Mathematics, vol. 21 (1969), pp. 11601164.
[18]Thompson, R.J., Embeddings into finitely generated simple groups which preserve the word problem, Word problems II (conf. on decision problems in algebra, Oxford, 1976), Studies in Logic and the Foundations of Mathematics, vol. 95, North-Holland, Amsterdam, New York, 1980, pp. 401–441.



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