Functional properties of limits of Sobolev homeomorphisms with integrable distortion

The functional and geometric properties of limits of homeomorphisms with integrable distortion of domains in Carnot groups are studied. The homeomorphisms belong to Sobolev classes. Conditions are obtained under which the limits of sequences of such homeomorphisms also belong to the Sobolev class, have a finite distortion, and have the $\mathcal N^{-1}$-Luzin property. In the case of Carnot groups of $H$-type, sufficient conditions are obtained that are imposed on domains and a sequence of homeomorphisms under which the limit mapping is injective almost everywhere. These results play a key role in finding extremal solutions to problems in the mathematical theory of elasticity on $H$-type Carnot groups, which are the subject of subsequent works by the authors.