The Geometry of Rewriting Systems: A Proof of the Anick-Groves-Squier Theorem

Abstract

Let G be a group or monoid which is presented by means of a complete rewriting system. Then one can use the resulting normal forms to collapse the classifying space of G down to a quotient complex (typically "small") of the same homotopy type. If the rewriting system is finite, then the quotient complex has only finitely many cells in each dimension. The proof yields an explicit free resolution of Z over ZG, similar to resolutions obtained by Anick, Groves, and Squier.