Here is a theorem of Hjorth-Kechris-Louveau which might be of some interest.

**Theorem**: Let $E^X_G$ be the orbit equivalence relation induced by a Borel action of a closed subgroup $G$ of $Sym(\mathbb{N})$ on some standard Borel space $X$. Then the following are equivalent:

i. $E^X_G$ is essentially countable.

ii. For some Polish topology $\tau$ on $X$ giving its Borel structure, $E^X_G$ is $\mathbf{\Sigma^0_2}$ in $(X,\tau)^2$.

ii. For some Polish topology $\tau$ on $X$ giving its Borel structure, $E^X_G$ is $\mathbf{\Sigma^0_3}$ in $(X,\tau)^2$.

(This is Theorem 1.12 in Jackson, Kechris and Louveau's paper "Countable Borel equivalence relations" and the following papers are given as references for this theorem: [Borel equivalence relations induced by actions of the symmetric group][1], [Borel equivalence relations and classifications of countable models][2])


  [1]: http://www.sciencedirect.com/science/article/pii/S0168007297000493
  [2]: http://www.sciencedirect.com/science/article/pii/S0168007296000061#