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Abstract

We investigate topological $\mathop {\rm AE}\nolimits (0)$-groups, a class which contains the class of Polish groups as well as the class of all locally compact groups. We establish the existence of a universal $\mathop {\rm AE}\nolimits (0)$-group of a given weight as well as the existence of a universal action of an $\mathop {\rm AE}\nolimits (0)$-group of a given weight on an $\mathop { \rm AE}\nolimits (0)$-space of the same weight. A complete characterization of closed subgroups of powers of the symmetric group $S_{\infty }$ is obtained. It is also shown that every $\mathop {\rm AE}\nolimits (0)$-group is Baire isomorphic to a product of Polish groups. These results are obtained by using the spectral descriptions of $\mathop { \rm AE}\nolimits (0)$-groups which are presented in Section~3.