Assume that $X$ is a compact metric space and $G$ is compact
 totally disconnected group. And $X$ has isometric free $G$-action
 i.e. $gx=x\Rightarrow g=e$. 



> 
> Then the following holds $${\rm dim}\ X/G={\rm dim}\ X-{\rm
>  dim}\ G\cdot o$$ where $o\in X$ and dimension is Hausdorff dimension ?
> 
> 
> 


Thank you in advance. 



Dimension of quotient of compact totally disconnected group action
It may happen that
$$\dim\_H X/G<\dim\_H X-\dim\_H (G\cdot o);$$
the following example is almost identical to Example 7.8 in "Fractal geometry" by Kenneth Falconer which provides spaces $X$ and $Y$ such that
$$\dim\_H (X\times Y)>\dim\_HX+\dim\_HY.$$


Denote by $W(\varepsilon)$ the two-point space with distance $\varepsilon$ between the pair of points.
Consider the $\ell\_\infty$-product 
$$X=\prod\_{n\in\mathbb{N}}W(\tfrac1{2^n}).$$
Fix a subset $\Sigma\in \mathbb{N}$ and consider the natural action of group
$$G=\prod\_{n\in\Sigma}\mathbb{Z}\_2$$
on $X$.


The orbit can be identified with the infinite product
$$G\cdot o=\prod\_{n\in\Sigma}W(\tfrac1{2^n}).$$
and the quotient space can be identified with the infinite product
$$X/G=\prod\_{n\notin\Sigma}W(\tfrac1{2^n}).$$


Note that for appropriate choice of $\Sigma$, we have
$$\dim (G\cdot o)= \dim X/G=0$$ 
while
$$\dim X=1.$$


