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Abstract

Let $G$ be a countably infinite group. We show that for every finite absolute coretract $S$, there is a regular left invariant topology on $G$ whose ultrafilter semigroup is isomorphic to $S$. As consequences we prove that (1) there is a right maximal idempotent in $\beta G\setminus G$ which is not strongly right maximal, and (2) for each combination of the properties of being extremally disconnected, irresolvable, and nodec, except for the combination $(-,-,+)$, there is a corresponding regular almost maximal left invariant topology on $G$.