Adian-Lisenok groups and (U) condition

A group G possesses the property (U) with respect to S if there exists a number M = M(G) such that for each generating set P of the group G there exists an element t ? G for which max _x?S |t ^?1 xt| _P ≤ M. It is proved that the well-known Adian-Lisenok groups possess the property (U). In connection with the problem on finding infinite groups with the property (U), which is stated in a joint unpublishedwork by D.Osin and D. Sonkin, it is shown that for any odd n ≥ 1003 there is a continuum set of non-isomorphic, i.e. simple groups with the property (U) in the variety of groups satisfying the identity x ⁿ = 1.