\documentclass[12pt]{amsart} \begin{document} \begin{center} Amenability, twisted inner amenability, and ICC\\ Evgenij Troitsky (joint work with Alexander Fel'shtyn and Nikita Luchnikov) \end{center} Let $\phi$ be an automorphism of a discrete group $G$. The talk is devoted to the study of the twisted inner representation $\gamma^\phi_G$ defined by $$ \gamma^\phi_G(x)(f)(g)=f(xg\phi(x^{-1})),\quad x,g\in G,\quad f\in\ell^2(G). $$ We prove under supposition of finiteness of stabilizers of $\phi$-twisted action, that $\gamma^\phi_G$ is weakly contained in the regular representation $\lambda_G$. Moreover, $\gamma^\phi_G$ is weakly contained in $\lambda_G$ if and only if the stabilizer $C_\phi (a)$ of the $\phi$-twisted action is amenable for all $a\in G$. It is proved that $\lambda_G$ is weakly contained in $\gamma^\phi_G$ for any ICC group $G$. Consider an automorphism $\phi$ of a finitely generated residually finite group $G$ with finite Reidemeister number. Then $G$ is $\phi$-inner amenable in an appropriate sense if and only if it is amenable. This differs from the case of inner amenability (i.e. $\mathrm{Id}$-inner amenability). \end{document}