In the board game Go, connected stones form groups whose survival depends on their liberties, the adjacent empty points (outer boundary). We determine the maximum number of liberties a group can achieve on an $N \times N$ board by establishing a useful duality between outer boundaries of groups and minimum connected dominating sets in grid graphs. Using previous results on the connected domination number of grids, we derive an explicit formula for all board sizes and prove that the maximum on the standard $19 \times 19$ board is exactly 229 liberties.
We provide explicit constructions for all $N$ and enumerate all optimal groups up to symmetry for $N \leq 7$. We also consider groups with special properties: non-branching, non-separating (doubly connected), and null-homotopic. In particular, we prove that all optimal groups are null-homotopic, that minimum connected dominating sets are path-shaped only for $N \in \{1,2,3,5,6\}$, and characterize when optimal groups are doubly connected.