# Project A4: Model theory, stable groups and corresponding complexes and geometries

## Principal investigator

## Participating scientists

- Dr. Martin Bays
- Javier de la Nuez Gonzalez
- Dr. Shelly Garion
- Dr. Zaniar Ghadernezhad
- Dr. Immanuel Halupczok
- apl. Prof. Dr. Lutz Hille
- Prof. Dr. Linus Kramer
- Isabel Mueller
- Dr. Lars Scheele

## Summary

We plan to further investigate the model theory of torsionfree hyperbolic groups and the corresponding complexes and to study and construct different examples of geometries arising from forking in stable and in particular omega-stable theories. This will be work in collaboration with R. Sklinos. This will require a better understanding of the 'basic' definable sets in the first order theories of these structures.In this context, the notion of 'ampleness' is a crucial concept to describe the stability properties of the underlying theory. The motivating example of an ample theory are projective spaces. It is well-known that the underlying field is definable in them. Work of Ould-Houcine and the PI shows that free and hyperbolic groups are also ample, but it is still not known whether an infinite field might be interpretable in these theories. The examples for ampleness lead to new questions about the model theory of the underlying curve complex, which will be studied in the project.

Recently the PI constructed the first examples of $\omega$-stable structures (of infinite Morley rank) which are ample, but do not interpret an infinite field. They are also the first examples that show that ampleness provides a proper hierarchy. Another aim of the project will be the construction of ample theories of finite Morley rank.

In another direction we also plan to investigate the model theory of profinite groups and valued fields. We hope to show that profinite NIP-groups are necessarily p-adic analytic, which would complement the work of D. Macpherson and the PI on pseudofinite groups. Also, in collaboration with U. Hartl and F. Jahnke we plan to use model theoretic methods to establish a notion of dimension for affine Deligne-Lusztig varieties in mixed characteristic in order to extend results from the equal characteristic setting.