Arboreal structures on Pruefer domains and applications
by
Serban A. Basarab
Institute of Mathematics of the Romanian Academy

Using algebraic and model theoretic methods a theorem on relative elimination of quantifiers for henselian valued fields of characteristic 0 is proved in [1], extending in this way wellknown results of MacIntyre and Prestel-Roquette on p-adically closed fields.A consequence of a purely algebraic nature-an isomorphism criterion for henselian valued fields algebraic over a common valued subfield is presented in the joint paper[2].In these works an essential role is played by some residue structures called mixed structures associated to a valued field which reflect the additive as well as multiplicative structures of the valued field.Roots of this approach can be traced in classical papers of Hasse and Krasner, and quite recently I learned that structures of this type occur in a special important case in a paper of Deligne[3].

Using a different point of view, in a work [4] devoted to generalized trees as compatible families of ßemiracks" I sketched the possibility of extending the Tits' construction of the tree of a valued field to Pruefer domains.Meanwhile I developed a systematic study of the generalized trees and their applications to the combinatorial group theory in the works [6-12].Recently I realized that the mixed structures from [1, 2]can be suitably generalized to Pruefer domains and in their investigation the methods of the arboreal group theory could play an important role.

The talk is devoted to the approach of the residue structures associated to a Pruefer domain with methods of the theory of generalized trees and to applications to the arboreal structure of SL2(K) where K is a global field and connections with the statistical theory of numbers[13, 14].

References.

1. S. Basarab, Relative elimination of quantifiers for Henselian valued fields, Annals Pure Appl.Logic 53 (1991) 51-74.
2. S. Basarab, F.V. Kuhlmann, An isomorphism theorem for algebraic extensions of valued fields, Manuscripta Math. 77 (1992) 113-126; 84 (1994) 113-114.
3. P. Deligne, Les corps locaux de caracteristique p, limites de corps locaux de caracteristique 0, dans "Representations des groupes reductifs sur un corps local. Travaux en cours, Paris Hermann (1984) 120-157.
4. S. Basarab, Generalized trees as compatible families of semiracks, communicated to York 1993 NATO Advanced Study Institute on Semigroups, Formal Languages and Groups.
5. J.P. Serre, "Trees", Springer, 1986.
6. S. Basarab, On a problem raised by Alperin and Bass, in R.C.Alperin (ed) Ärboreal Group Theory", Springer (1991) 35-68.
7. S. Basarab, On a problem raised by Alperin and Bass, 1:Group actions on groupoids, J.Pure Appl.Algebra 73 (1991) 1-12.
8. S. Basarab, The dual of the category of trees, Preprint Series IMAR nr.7 (1992).
9. S. Basarab, On a problem raised by Alperin and Bass, 2:Metric and order theoretic aspects, Preprint Series IMAR, nr.10 (1992).
10. S. Basarab, Directions and foldings on generalized trees, Fundamenta Informaticae, 30:2 (1997) 125-149.
11. S. Basarab, On discrete hyperbolic arboreal groups, Communications in Algebra, 26:9 (1998) 2837-2866.
12. S. Basarab, Partially commutative Artin-Coxeter groups and their arboreal structure, 1-3, Preprint Series IMAR, nr.5, 7, 11 (1997).
13. J.B. Bost, A.Connes, Hecke algebras, type 3 factors and phase transitions with spontaneous symmetry breaking in number theory, IHES Preprint 1995.
14. D.Harari, E.Leichtnam, Extension du phenomene de brisure spontanee de symetrie de Bost-Connes au cas des corps globaux quelconques, Preprint 1996.

Date received: April 28, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacv-30.