# On first order amenability

@article{Hrushovski2020OnFO, title={On first order amenability}, author={Ehud Hrushovski and Krzysztof Krupinski and Anand Pillay}, journal={arXiv: Logic}, year={2020} }

We introduce the notion of first order [extreme] amenability, as a property of a first order theory $T$: every complete type over $\emptyset$, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure [type] in the same variables. [Extreme] amenability of $T$ will follow from [extreme] amenability of the (topological) group $Aut(M)$ for all sufficiently large $\aleph_{0}$-homogeneous countable models $M$ of $T$ (assuming $T$ to be countable), but is…

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## References

SHOWING 1-10 OF 30 REFERENCES

### Topological dynamics and the complexity of strong types

- MathematicsIsrael Journal of Mathematics
- 2018

We develop topological dynamics for the group of automorphisms of a monster model of any given theory. In particular, we find strong relationships between objects from topological dynamics (such as…

### On NIP and invariant measures

- Mathematics
- 2007

We study forking, Lascar strong types, Keisler measures and definable groups, under an assumption of NIP (not the independence property), continuing aspects of the paper [13]. Among key results are…

### Stable group theory and approximate subgroups

- Mathematics
- 2009

We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group G, we show that a finite…

### Tits Buildings and the Model Theory of Groups: Introduction to the Lascar Group

- Mathematics
- 2002

The aim of this article is to give a short introduction to the Lascar Galois group GalL(T ) of a complete first order theory T . We prove that GalL(T ) is a quasicompact topological group in section…

### Hyperimaginaries and automorphism groups

- MathematicsJournal of Symbolic Logic
- 2001

This paper shows that if T is simple and canonical bases of Lascar strong types exist in Meq then hyperimaginaries can be eliminated in favour of sequences of ordinary imaginaries, and develops a Galois theory of T, making use of the structure of compact groups.

### The diameter of a Lascar strong type

- Mathematics
- 2003

We prove that a type-denable Lascar strong type has nite diameter. We also answer some other questions from (1) on Lascar strong types. We give some applications on subgroups of type-denable groups.…

### Boundedness and absoluteness of some dynamical invariants in model theory

- MathematicsJ. Math. Log.
- 2019

It is shown that in each of these two cases, boundedness of a minimal left ideal is an absolute property and that whenever such an ideal is bounded, then its isomorphism type is also absolute.

### Groups, measures, and the NIP

- Mathematics
- 2006

We discuss measures, invariant measures on definable groups, and genericity, often in an NIP (failure of the independence property) environment. We complete the proof of the third author’s…