# Nilpotent Group-Counterexamples to Zilbers Conjecture

## TL;DR

This paper constructs specific 3-nilpotent groups that serve as counterexamples to Zilbers Conjecture, using graded Lie algebras over finite fields, expanding the understanding of model-theoretic properties of such groups.

## Contribution

It introduces uncountably categorical 3-nilpotent groups of exponent p > 3 that are counterexamples to Zilbers Conjecture, employing the additive Collapse method.

## Key findings

- Constructed uncountably categorical 3-nilpotent groups of exponent p > 3
- Demonstrated these groups are not one-based and do not interpret infinite fields
- Extended previous work on 2-nilpotent groups to 3-nilpotent case

## Abstract

We construct uncountably categorical 3-nilpotent groups of exponent p > 3. They are not one-based and do not allow the interpretation of an infinite field. Therefore they are counterexamples to Zilbers Conjecture. First 2-nilpotent new uncoutably categorical groups were contructed in [3]. Here we use the method of the additive Collapse developed in [5]. Essentially we work with 3-nilpotent graded Lie algebras over the field with p elements.