Quasi-isometric Higman embeddings and the Dehn function

Abstract

This is the first of a sequence of papers devoted to studying the link between the complexity of the Word Problem for a finitely generated recursively presented group $G$ and the isoperimetric functions of the finitely presented groups in which $G$ embeds. We prove here that if a finitely generated group has a presentation $\mathcal{P}$ whose relators can be enumerated by a computational model satisfying certain technical requirements, then the group embeds quasi-isometrically into a finitely presented group whose Dehn function is bounded above by a function of the model's computational complexity and the Dehn function of $\mathcal{P}$. This generalizes a previous result of the author pertaining to the embeddings of free Burnside groups and gives a recipe for establishing such Higman embeddings into groups with desired geometric properties. As an example of the use of this embedding scheme, we find a substantial improvement to the seminal result of Birget, Ol'shanskii, Rips, and Sapir showing that the Word Problem of a finitely generated group is in class NP if and only if the group embeds into a finitely presented group with polynomial Dehn function.