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We introduce a notion of intrinsically Lipschitz graphs in the context ofmetric spaces. This is a broad generalization of what in Carnot groups has beenconsidered by Franchi, Serapioni, and Serra Cassano, and later by many others.We proceed by focusing our attention on the graphs as subsets of a metric spacegiven by the image of a section of a quotient map and we require anintrinsically Lipschitz condition. We shall not have any function on atopological product, not we shall consider a metric on the base of the quotientmap. Our results are: an Ascoli-Arzel\`a compactness theorem, an Ahlforsregularity theorem, and some extension theorems for partially definedintrinsically Lipschitz sections. Known results by Franchi, Serapioni, andSerra Cassano, and by Vittone will be our corollaries.

Intrinsically Lipschitz sections and applications to metric groups