Core surfaces

Let $$\Gamma _{g}$$ be the fundamental group of a closed connected orientable surface genus $$g\ge 2$$ . We introduce combinatorial structure core surfaces, that represent subgroups These structures are (usually) 2-dimensional complexes, made up vertices, labeled oriented edges, and 4g-gons. They compact whenever corresponding subgroup is finitely generated. The theory surfaces we initiate here analogous to influential fruitful Stallings graphs for free groups.