Elements with trivial centralizer in wreath products

Groups with self-centralizing elements have been investigated in recent papers by Kappe, Konvisser and Seksenbaev. In particular, if $G = A$wr$B$ is a wreath product some necessary and some sufficient conditions have been given for the existence of self-centralizing elements and for $G = \left\langle {{S_G}} \right\rangle$, where ${S_G}$ is the set of self-centralizing elements. In this paper ${S_G}$ and the set ${R_G}$ of elements with trivial centralizer are determined both for restricted and unrestricted wreath products. Based on this the size of $\left\langle {{S_G}} \right\rangle$ and $\left\langle {{R_G}} \right\rangle$ is found in some cases, in particular if $A$ and $B$ are $p$-groups or if $B$ is not periodic.