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.
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.
