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A horizontal k-polygonal chain on a Carnot group is the union of a finite number k of segments Ii, i = 1, ..., k, of integral lines of horizontal vector fields that are linear combinations of basic left-invariant horizontal vector fields of the Carnot group under consideration (links of a horizontal polygonal chain). In this case, the end of the link Ii is the beginning of the link Ii + 1 for i = 1, ..., k-1. It is known that for every Carnot group G there is a number KG such that any two points of G can be connected by a horizontal k-polygonal chain, where k ≤ KG. Our talk is devoted to finding the minimum number KG or estimate it on specific 2- and 3-step Carnot groups.