Group construction in non-trivial geometric $C$-minimal structures

TL;DR

This paper proves that in certain complex geometric structures, any infinite group can be explicitly defined, expanding understanding of their algebraic properties without assuming linearity.

Contribution

It demonstrates that infinite groups are definable in non-trivial geometric $C$-minimal structures under specific conditions, without assuming linearity.

Findings

Infinite groups are definable in these structures.

No linearity assumptions are needed.

Conditions include being definably maximal and lacking certain bijections.

Abstract

We show that an infinite group is definable in any non trivial geometric $C$-minimal structure which is definably maximal and does not have any definable bijection between a bounded interval and an unbounded one in its canonical tree. No kind of linearity is assumed.