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If $S $ is infinite, locally finite graph which is not tree, $\tilde{S} $ is its universal cover, $p:\tilde{S}\rightarrow S $ is covering map,and $G $ is acting on $\tilde{S} $ with finite quotient $Y $, $q:\tilde{S}\rightarrow Y $ is the corresponding covering, can we find a group $H $ which acts on $S $ with quotient $Y$ such that the diagram commutes (i.e. $r\circ p=q $ where $r:S\rightarrow Y=S/H $ is natural map)?

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    @Theo: Here I have assumed $S$ is infinite graph which is not tree and $Y$ is a ***finite*** quotient of $\tilde{S}$. therefore, I was expecting whether there is such map from $S$ to $Y$.2011-08-29

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