Mekler’s construction!

Question

I was looking at this slides by Artem Chernikov. But I did not understad what Mekler’s construction is exactly.

Can one explain the idea of Mekler’s construction (in model theory) in a simple words? Thank you!

Answer 1

The best place to read about Mekler's construction is Hodges's big book Model Theory.

The construction proceeds in two steps:

1. From a structure $A$ in a finite relational language, produce a "nice" graph $C_A$ (here "nice" has a precise definition) which is bi-interpretable with $A$. Hodges works this out in detail in Theorem 5.5.1.
2. From a "nice" graph $C$, look at the nilpotent class 2 group of exponent $p$ (for a fixed prime $p$) $G(C)$ presented by generators and relations, where the generators are the vertices of $C$ and the relations say that two generators commute if and only if they are connected by an edge in $C$. So explicitly, we take the free group generated by $C$ and quotient by the normal subgroup generated by (a) $x^p$ for all $x$, (b) $[[x,y],z]$ for all $x,y,z$, (c) $[a,b]$ for all generators $a,b\in C$ which are connected by an edge in $C$. Here $[x,y]$ is notation for the commutator $xyx^{-1}y^{-1}$.

The analysis of this group $G(C)$, showing that it reflects the model theoretic properties of $C$ (and hence of $T$ if $C = C_A$ where $A\models T$) is carried out in Appendix A.3 of Hodges.

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In the comments, you ask for a "simple example". The problem is that even if you start with a very simple theory $T$, stage 1 of the above construction produces a big ugly graph. And then the definition of $G(C)$ in terms of generators and relations in stage 2 is as concrete a description of this group as you're going to get. So there's not much more to say about examples.

If you're having trouble with the abstractness of the definition of $G(C)$, it could be instructive to take a small finite "nice" graph $C$ (like the one which is just two points connected by an edge) and try to understand the group $G(C)$. Similarly, you could try to understand stage 1 by taking a small finite structure $A$ in a language with just one or two relation symbols, and drawing the picture of $C_A$.