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Let $U$ be a complex analytic space with an action of a finitely generated group $\Gamma$. Under what assumptions is the following true:

Every $\Gamma$-invariant closed analytic subset of $U$ decomposes into a finite union $U=U_1\cup ...\cup U_n$ of closed analytic sets such that connected components of these sets $U_i, 1\leq i\leq n$ are irreducible. And if one further requires that each of the $U_i$'s is fixed by a finite index subgroup of $\Gamma$?

A case of interest is when $U$ a universal covering space of a complex algebraic variety and $\Gamma$ is its fundamental group. Is this necessarily true if the fundamental group is residually finite? And if one further assumes that $U$ is Stein ?

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  • $\begingroup$ For $U$ upper halfplane and $\Gamma$ Fuchsian, can you give an example? I don't see it. $\endgroup$
    – Marc Palm
    Commented Aug 27, 2013 at 15:29
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    $\begingroup$ @MarcPalm: In this case, any proper analytic subset $A\subset U$ (one just should not denote it $U$ as OP does) is discrete and the decomposition is $A=A_1$: All connected components of $A$ are points, and, hence, irreducible. $\endgroup$
    – Misha
    Commented Aug 28, 2013 at 5:25

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