If is a compact surface with boundary, then a finitely generated subgroup without peripheral elements of can be separated from finitely many other elements of by a finite index subgroup of corresponding to a finite cover with the same number of boundary components as .
"Conservative subgroup separability for surfaces with boundary." Algebr. Geom. Topol. 13 (1) 115 - 125, 2013. https://doi.org/10.2140/agt.2013.13.115