Groups definable in tame expansions of o-minimal structures and related problems.
Final Report Abstract
This project concerned work in and around o-minimality, focusing on the study of definable groups. Examples of groups definable in an o-minimal structure include all compact real Lie groups. The project included 6 research goals divided into two parts. The first part (Goals 1 - 3) focused on definable groups in tame expansions M = R, P of an o-minimal structure R studied here for the first time. Examples of such expansions include M = R, P , where R = R is the real field, and P = Qrc is the field of real algebraic numbers, or the multiplicative group P = 2Q of rational powers of 2, or a transcendence basis over Q. In these settings, we obtained the following key results: (a) a decomposition theorem for all definable sets in M in terms of their R-definable and P -internal ‘parts’, (b) the development of dimension theory for M, (c) a local theorem for definable groups in M, and (d) a global theorem for definable groups in the special case when P is an independent set. Based on these results, Goals 1 and 3 were completed, whereas Goal 2 was completed by another author by the time this work began. The second part (Goals 4 - 6) concerned related problems in the o-minimal setting. The key findings include: (a) the solution of an important instance of a conjecture on locally definable groups, stated by the PI and Peterzil in 2012, (b) the development of algebraic topology machinery in the semilinear setting towards applications in Hrushovski-Loeser spaces, (c) the solution of the Zilber Dichotomy Conjecture in the o-minimal setting, stated by Peterzil in 2005. Based on (c), Goal 6 was completed. Based on (a), the conjecture in Goal 4 was answered in one important instance. The result in (b) replaced the original questions in Goal 5 (again on semilinear algebraic topology) towards better matching with concurrent developments in Hrushovski-Loeser spaces. The results of the first part and their methodology has strong potential to be used in similar analysis of definable groups in more general modeltheoretic settings, such as NIP (not the independence property). By-products of the current work include the first point counting theorems in tame expansions of o-minimal structures, extending the influential Pila-Wilkie theorem from the ominimal setting to the general tame setting. The results of the second part open the way to further applications, such as to the Hrushovski-Loeser spaces, as well as to Diophantine applications based on the Zilber Dichotomy in the o-minimal setting.
Publications
- Semi-linear stars are contractible, Fund. Math. 241 (2018), 291–312
P. Eleftheriou
(See online at https://doi.org/10.4064/fm394-10-2017) - Characterizing o-minimal groups in tame expansions of ominimal structures, J. Inst. Math. Jussieu
P. Eleftheriou
(See online at https://doi.org/10.1017/S1474748019000392) - Locally definable subgroups of semialgebraic groups, J. Math. Logic
E. Baro, P. Eleftheriou, Y. Peterzil
(See online at https://doi.org/10.1142/S0219061320500099) - Small sets in dense pairs, Israel J. Math.
P. Eleftheriou
(See online at https://doi.org/10.1007/s11856-019-1892-4) - Strongly minimal groups in o-minimal structures, J. Eur. Math. Soc.
P. Eleftheriou, A. Hasson, Y. Peterzil
- Structure theorems in tame expansions of o-minimal structures with a dense predicate, Israel J. Math.
P. Eleftheriou, A. Günaydin, P. Hieronymi
