Injectivity theorems and algebraic closures of groups with coefficients

摘要

Recently, Cochran and Harvey defined torsion-free derived series of groupsand proved an injectivity theorem on the associated torsion-free quotients. Weshow that there is a universal construction which extends such an injectivitytheorem to an isomorphism theorem. Our result relates injectivity theorems to acertain homology localization of groups. In order to give a concretecombinatorial description and existence proof of the necessary homologylocalization, we introduce a new version of algebraic closures of groups withcoefficients by considering a certain type of equations.