Let U be a word in letters x1±1, . . . , xm±1, m > 2, and a group G be given by presentation G = 〈x, . . . , xm||UxiU-1 = Xi+1, i = 1, . . . , m -1〉. It is proven that this presentation is aspherical provided the word U does not have the form U2U1, where U1 is a word in letters x1±1, . . . , xm-1±1 and U2 is a word in letters x2±1, . . . , m±1. It is also proven that the (images of) x1, . . . , xm-1 freely generate a free subgroup of G if and only if the word U does not have the foregoing form U2U1.
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