Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-19T12:35:03.341Z Has data issue: false hasContentIssue false
  • English
  • Français

Interprétation de l'arithmétique dans certains groupes de permutations affines par morceaux d'un intervalle

Part of: Model theory Special aspects of infinite or finite groups Computability and recursion theory

Published online by Cambridge University Press:  30 January 2009

Tuna Altinel  and
Alexey Muranov
Tuna Altinel
Affiliation:
Université de Lyon, Université Lyon 1, Institut Camille Jordan CNRS UMR 5208, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France ([email protected])
Alexey Muranov
Affiliation:
Université de Lyon, Université Lyon 1, Institut Camille Jordan CNRS UMR 5208, 43, boulevard du 11 novembre 1918, F-69622 Villeurbanne Cedex, France ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

The arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the elementary theories of all these groups are undecidable. Moreover, Thompson's group F and some of its generalizations interpret the arithmetic without parameters.

Résumé

L'arithmétique est interprétée dans tous les groupes de Richard Thompson et de Graham Higman, aussi bien que dans d'autres groupes des permutations affines par morceaux d'un intervalle qui généralisent les groupes de Thompson et de Higman. En particulier, les théories élémentaires de tous ces groupes sont indécidables. De plus, le groupe F de Thompson et certaines de ses généralisations interprètent l'arithmétique sans paramètres.

Keywords

groupes de Thompsongroupe simple de présentation finiethéorie élémentaireinterprétationarithmétiqueindécidabilité héréditaireThompson groupsfinitely presented simple groupselementary theoryinterpretationarithmetichereditary undecidability

MSC classification

Secondary: 03C62: Models of arithmetic and set theory 20F65: Geometric group theory 03D35: Undecidability and degrees of sets of sentences
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 8 , Issue 4 , October 2009 , pp. 623 - 652
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

1.Ahlbrandt, G. et Ziegler, M., Quasi finitely axiomatizable totally categorical theories, Annals Pure Appl. Logic 30(1) (1986), 63–82.CrossRefGoogle Scholar
2.Bardakov, V. G. et Tolstykh, V. A., Interpreting the arithmetic in Thompson's group F, J. Pure Appl. Alg. 211(3) (2007), 633–637.CrossRefGoogle Scholar
3.Belk, J. M. et Brown, K. S., Forest diagrams for elements of Thompson's group F, Int. J. Alg. Comput. 15 (2005), 815–850.CrossRefGoogle Scholar
4.Bieri, R. et Strebel, R., On groups of PL-homeomorphisms of the real line, notes non publiées 1985.Google Scholar
5.Bleak, C., Gordon, A., Graham, G., Hughes, J., Matucci, F., Newfield-Plunkett, H. et Sapir, E., Using dynamics to analyze centralizers in the generalized Higman–Thompson groups Vn, prépublication en version incomplète (2007).Google Scholar
6.Brin, M. G. et Squier, C. C., Presentations, conjugacy, roots, and centralizers in groups of piecewise linear homeomorphisms of the real line, Commun. Alg. 29(10) (2001), 4557–4596.CrossRefGoogle Scholar
7.Brown, K. S., Finiteness properties of groups, J. Pure Appl. Alg. 44 (1987), 45–75.CrossRefGoogle Scholar
8.Cannon, J. W., Floyd, W. J. et Parry, W. R., Introductory notes on Richard Thompson's groups, Enseign. Math. 42 (1996), 215–256.Google Scholar
9.Davis, M., Hilbert's tenth problem is unsolvable, Am. Math. Mon. 80 (1973), 233–269.CrossRefGoogle Scholar
10.Delon, F. et Simonetta, P., Undecidable wreath products and skew power series fields, J. Symb. Logic 63(1) (1998), 237–246.CrossRefGoogle Scholar
11.Dennis, R. K. et Vaserstein, L. N., Commutators in linear groups, K-Theory 2(6) (1989), 761–767.CrossRefGoogle Scholar
12.Ershov, Y. L., Problemy razreshimosti i konstruktivnye modeli [« Problèmes de décidabilité et modèles constructifs »], Matematicheskaya Logika i Osnovaniya Matematiki (Nauka, Moscow, 1980; en russe).Google Scholar
13.Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I [«Sur les propositions formellement indécidables des Principia Mathematica et des systèmes apparentés, I»], Monatsh. Math. 149(1) (2006), 1–30 (en allemand) (réimprimé avec une introduction par S.-D. Friedman sur Monatsh. Math. Phys. 38 (1931), 173–198).CrossRefGoogle Scholar
14.Higman, G., Finitely presented infinite simple groups, Notes on Pure Mathematics, Volume 8 (Australian National University, Canberra, 1974).Google Scholar
15.Hodges, W., Model theory, Encyclopedia of Mathematics and Its Applications, Volume 42 (Cambridge University Press, 1993).CrossRefGoogle Scholar
16.Hodges, W., A shorter model theory (Cambridge University Press, 1997).Google Scholar
17.Khélif, A., Biinterprétabilitée et structures QFA : étude de groupes résolubles et des anneaux commutatifs, C. R. Acad. Sci. Paris Sér. I 345(2) (2007), 59–61.CrossRefGoogle Scholar
18.Matijasevich, Y. V., Diofantovost' perechislimykh mnozhestv [« La nature diophantienne des ensembles énumérables»], Dokl. Akad. Nauk SSSR 191 (1970), 279–282 (en russe; traduction anglaise dans Sov. Math. Dokl.).Google Scholar
19.Mostowski, A. et Tarski, A., Undecidability in the arithmetic of integers and in the theory of rings, J. Symb. Logic 14 (1949), 76.Google Scholar
20.Mostowski, A., Robinson, R. M. et Tarski, A., Undecidability and essential undecidability in arithmetic, in Undecidable theories: studies in logic and the foundations of mathematics, pp. 36–74 (North-Holland, Amsterdam, 1971).Google Scholar
21.Nies, A., Describing groups, Bull. Symb. Logic 13(3) (2007), 305–339.CrossRefGoogle Scholar
22.Noskov, G. A., Ob elementarnoj teorii konechno porozhdyennoj pochti razreshimoj gruppy [«À propos de la théorie élémentaire d'un groupe virtuellement résoluble de type fini»], Izv. Akad. Nauk SSSR Ser. Mat. 47(3) (1983), 498–517 (en russe; traduction anglaise dans Math. USSR Izv.).Google Scholar
23.Noskov, G. A., On the elementary theory of a finitely generated almost solvable group, Math. USSR Izv. 22(3) (1984), 465–482.CrossRefGoogle Scholar
24.Pillay, A., An introduction to stability theory, Oxford Logic Guides, Volume 8 (Clarendon/Oxford University Press, New York, 1983).Google Scholar
25.Poizat, B., Cours de théorie des modèles: une introduction à la logique mathématique contemporaine (Nur al-Mantiq wal-Ma'rifah, Bruno Poizat, Lyon, 1985).Google Scholar
26.Poizat, B., Groupes stables: une tentative de conciliation entre la géométrie algébrique et la logique mathématique (Nur al-Mantiq wal-Ma'rifah, Volume 2, Bruno Poizat, Lyon, 1987).Google Scholar
27.Poizat, B., Stable groups (traduit de l'original françcais de 1987 par M. G. Klein), Mathematical Surveys and Monographs, Volume 87 (American Mathematical Society, Providence, RI, 2001).CrossRefGoogle Scholar
28.Prest, M. Y., Model theory and modules, London Mathematical Society Lecture Notes, Volume 130 (Cambridge University Press, 1988).CrossRefGoogle Scholar
29.Robinson, R. M., Undecidable rings, Trans. Am. Math. Soc. 70 (1951), 137–159.CrossRefGoogle Scholar
30.Rothmaler, P., Introduction to model theory (traduit et révisé de l'original allemand de 1995 par l'auteur), Algebra, Logic and Applications, Volume 15 (Gordon and Breach, Amsterdam, 2000).Google Scholar
31.Scanlon, T., Infinite finitely generated fields are biinterpretable with ℕ, J. Am. Math. Soc. 21(3) (2008), 893–908.CrossRefGoogle Scholar
32.Sela, Z., Diophantine geometry over groups VIII: stability, prépublication (http://arxiv.org/abs/math/0609096, 2006).Google Scholar
33.Stein, M., Groups of piecewise linear homeomorphisms, Trans. Am. Math. Soc. 332(2) (1992), 477–514.CrossRefGoogle Scholar
34.Tarski, A., A general method in proofs of undecidability, in Undecidable theories: studies in logic and the foundations of mathematics, pp. 1–35 (North-Holland, Amsterdam, 1971).Google Scholar
35.Wagner, F. O., Stable groups, in Handbook of algebra, Volume 2 (North-Holland, Amsterdam, 2000).Google Scholar