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I need to know the definition of an irregular (branched) cover. I heard this somewhere but I am not able to find any definition on the internet.

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According to this text, branced covering $p:Y\to X$ is called irregular, iff the covering $(Y/\textrm{Aut}\,p)\to X$ is not homeomorphism. Here $\textrm{Aut}\,p$ is the group of all homeomorphisms $Y\to Y$ over $X$. In other words, automorphisms group acts non-transitive on the fiber of $p$.

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