Algebra Seminar talk

Meng-Che Ho (Purdue University)
Finitely Generated Groups Are Universal

It is well-known that any structure can be coded in a graph. Hirschfeldt, Khoussainov, Shore, and Slinko showed that any structure can be coded in a partial ordering, lattice, integral domain, or 2-step nilpotent group, and showed that certain computability properties are preserved under this coding.

Recently, Miller, Poonen, Schoutens, and Shlapentokh added fields to this list, and they gave a category-theoretic framework to establish their result. Around the same time, Montalbán introduced the notion of effective interpretation and showed that effectively bi-interpretable structures share many computability-theoretic properties. Thus, using either of these notions, we may define the notions of a class of structures being universal. Harrison-Trainor, Melnikov, Miller, and Montalbán showed that these two notions are indeed equivalent.

In joint work with Harrison-Trainor, we use these ideas to define a class of structures being universal within finitely-generated structures. We then use small cancellation techniques from group theory to show that the class of finitely-generated groups with finitely-many constants named is universal within finitely-generated structures. One application of this is to the study of Scott sentences of finitely-generated groups. Knight et al. showed that a finitely-generated structure always has a $\Sigma_3$ Scott sentence, but most interesting classes of finitely-generated groups admit d-$\Sigma_2$ Scott sentences. Using the same coding technique, Harrison-Trainor and I construct a finitely-generated group that admits no d-$\Sigma_2$ Scott sentence.