# Mekler's construction and tree properties

## TL;DR

This paper explores how Mekler's construction, which transforms structures into pure groups, preserves various complex model-theoretic properties including NTP$_1$, NSOP$_1$, and NSOP$_2$, expanding understanding of these properties in group theories.

## Contribution

The paper demonstrates that Mekler's construction preserves NTP$_1$, NSOP$_1$, and NSOP$_2$, providing new insights into the transfer of model-theoretic properties to pure groups.

## Key findings

- Mekler's construction preserves NTP$_1$ and NSOP$_1$ properties.
- The construction also preserves NSOP$_2$, expanding its applicability.
- Results imply existence of pure group theories with complex properties derived from non-simple theories.

## Abstract

Mekler constructed a way to produce a pure group from any given structure where the construction preserves $\kappa$-stability for any cardinal $\kappa$. Not only the stability, it is known that his construction preserves various model-theoretic properties such as simplicity, NIP, and NTP$_2$. Inspired by the last result, we show that the construction also preserves NTP$_1$(NSOP$_2$) and NSOP$_1$. As a corollary, we obtain that if there is a theory of finite language which is non-simple NSOP$_1$, or which is NSOP$_2$ but has SOP$_1$, then there is a pure group theory with the same properties, respectively.