Sufficiency of broken lines in the modulus method and removable sets

We establish the sufficiency of the family of broken lines in calculating the modulus of a condenser. We extend the Ahlfors–Beurling definition of removable sets basing on rectangles to weighted Sobolev spaces with a Muckenhoupt weight. We obtain exact characteristics of removable sets in terms of girth by broken lines. We prove the invariance of weighted Sobolev spaces under quasi-isometric mappings.