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Among its definitions, dic.com gives:

The extreme edge or margin; a border. See Synonyms at border.
An enclosing boundary;

The space enclosed by such a boundary;

The point beyond which an action, state, or condition is likely to begin or occur; the brink: on the verge of tears; a nation on the verge of economic prosperity; and

A rod, wand, or staff carried as an emblem of authority or office.
Obsolete. The rod held by a feudal tenant while swearing fealty to a lord.

All have to do with domain and, therefore, boundaries: whether of property of power. The rod in bold is the symbol of authority over real property which was, in Europe, explicitly defined in terms of boundaries. Verge, in its original sense, is thus the symbol of boundary.