THE ASPHERICITY AND FREIHEITSSATZ FOR CERTAIN LOT-PRESENTATIONS OF GROUPS

Abstract

Let U be a word in letters [Formula: see text], m >2, and a group G be given by presentation G=< x1,…, xm‖ U xi U-1= xi+1, i=1,…, m-1>. It is proven that this presentation is aspherical provided the word U does not have the form U2 U1, where U1 is a word in letters [Formula: see text] and U2 is a word in letters [Formula: see text]. 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 U2 U1.