Coordinate recognition: General theory, Groups, and other surprises

A class of structures \emph{recognizes coordinates} if any reduced product of structures from said class witnesses a certain kind of rigidity phenomenon.

We provide several equivalent characterizations of this property.

This property has (at least) two remarkable consequences, one set-theoretic and one model-theoretic, for reduced products of structures of the said class.

First, under appropriate set-theoretic assumptions every isomorphism between such reduced products associated with the Fréchet ideal lifts (modulo a finite change) to an isomorphism between products of the original structures.

Second, with an additional mild assumption, it implies a strong quantifier elimination result.

Of note, we show that a class recognizes coordinates if and only if an individual formula witnesses a certain syntactic property.

We also consider many concrete classes of structures and determine whether or not they recognize coordinates.

We place heavy emphasis on well-known classes of groups, such as permutation groups, acylindircally hyperbolic groups, quasisimple groups, free products, and graph products, but we also discuss other classes of structures.