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.
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 $$\dim X=1.$$
