If a group is q.i. to $\mathbb Z$, then all its asymptotic cones are lines. In our paper with Cornelia Drutu, "Tree-graded spaces and asymptotic cones of groups", Topology 44 (2005), no. 5, 959--1058 (see Corollary 6.2), we prove that if even one asymptotic cone of a group is a line, then the group is virtually cyclic.
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