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We prove that any word hyperbolic group which is virtually compact special(in the sense of Haglund and Wise) is conjugacy separable. As a consequence wededuce that all word hyperbolic Coxeter groups and many classical smallcancellation groups are conjugacy separable.  To get the main result we establish a new criterion for showing that elementsof prime order are conjugacy distinguished. This criterion is of independentinterest; its proof is based on a combination of discrete and profinite(co)homology theories.

Virtually compact special hyperbolic groups are conjugacy separable