Distortion functions and the membership problem for submonoids of groups and monoids

The notion of upper distortion for graded submonoids embedded in groups and monoids is introduced. A finitely generated monoid M is graded if every element of M can be written in only finitely many ways in terms of some fixed system of generators. Examples of such monoids are free monoids, Artin monoids, and monoids satisfying certain small cancellation conditions. Whenever such a monoid is embedded in another monoid or a group G, we define an upper distortion function comparing the intrinsic word metric on M with the extrinsic word metric on M inherited from G. If the word problem in G is solvable, then the membership problem for M in G is solvable if and only if there exists a recursive upper distortion function for M in G. A particularly good aspect of upper distortion functions, when they exist, is that they lift well under homomorphisms. In order to illustrate this general approach, we solve the membership problem in positively generated submonoids of some one-relator groups, including Baumslag-Solitar groups, surface groups, and some groups given by Adian type relations. To solve these problems we use linear representations (quite often not faithful) in which long products of matrices have large matrix norms, construct upper distortion functions, and lift them back to the original groups.