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Given any space $X$ of Lebesgue dimension at most $n$. Suppose a group $G$ acts on $X$ continuously. Can the dimension of the quotient $G\backslash X$ exceed the dimension of $X$?

I know examples, where quotient maps increase the dimension. But I don't know an example, where this quotient is given by dividing out a group action.

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If you don't demand it's a topological group action don't your examples generalize naturally? For example, take a space-filling curve $[0,1] \to [0,1]^2$. There is an equivalence relation on $[0,1]$ and we ask is it induced by a group action? Each equivalence class is a set, with at most the cardinality of the continuum. So the symmetric group of the continuum acts transitively on each class. – Ryan Budney Oct 4 '10 at 20:10
Indeed. But I wanted the group action to be continuous. Otherwise there is also the following example: Take the cantor set equipped with the following $\mathbb{Z}/2$ -action: Let the nontrivial element act by exchanging the left and right endpoints of all removed intervals (and leaving all other points fixed). Of course this map is not continuous. The quotient is homeomorphic to $[0;1]$. – HenrikRüping Oct 4 '10 at 20:23
up vote 8 down vote accepted

There are examples of dimension-raising orbit maps arising from actions of p-adic groups. Quoting from the MathSciNet review of the following paper: Raymond, Frank; Williams, R. F. Examples of $p$-adic transformation groups. Ann. of Math. (2) 78 1963 92--106, review by P. Conner: "The authors of this paper show that if $A_p$, $p$ prime, is the compact 0-dimensional $p$-adic group, then for any integer $n\geq 2$ there is a compact $n$-dimensional metric space $X$ together with an action of $A_p$ upon $X$ as a group of transformations so that the dimension of the quotient space, $X/A_p$, is $n+2$."

It is unknown whether a $p$-adic group can act effectively on an $n$-manifold. It is known that if such an action exists, then the orbit space necessarily has higher dimension. The conjecture that no such action exists is known as the Hilbert-Smith Conjecture.

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Just as a remark. There is a reference in this paper to the short paper A.Kolmogoroff, "Über offene Abbildungen", which constructs a easier example for dim$(X/A_2)=n+1$, but which has the flaw to be in German. – HenrikRüping Oct 6 '10 at 14:31

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