Mekler's construction and tree properties

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.

11 pages