Identities in Algebras and Pro-P Groups

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One of the fundamental identities satisfied by number systems is the commutative law ab=ba. More complex algebraic systems such as sets of functions or symmetries can have much more intricate behavior, reflected by more complicated algebraic identities. This research project addresses questions such as whether the set of all such identities can be deduced from a finite subset. The results are expected to have applications in algebra, number theory, and theoretical computer science. This project aims to develop a new theory of infinite groups satisfying pro-unipotent or pro-p identities. The theory should be parallel to the theory of algebras with polynomial identities and provide a new approach to linear groups over commutative rings. Important steps in implementation of this program will be (1) a proof that linear pro-p and pro-unipotent groups satisfy nontrivial identities, (2) proofs that these identities are finitely based, and (3) a structure theory that would establish that under certain conditions groups with identities are linear.