Cohomology of group theoretic Dehn fillings I: Cohen-Lyndon type theorems

This is the first paper of three papers in a row aiming to study cohomology of group theoretic Dehn fillings. In the present paper, we prove a particular free product structure, which is termed the Cohen-Lyndon property, of Dehn filling kernels. As an application, we describe the structure of relative relation modules of Dehn fillings.