On the structure and extendibility of k-power free words

The branching of the tree of k-power free words is explored. It is shown that the set of k-power free ω-words over Σ is perfect if k > 6 and |Σ| > 1, if k > 3 and |Σ| > 2, if k > 2 and |Σ| > 3, or if k > 1 and |Σ| > 4. An effective way of showing when a finite k-power free word w over Σ can be extended to a k-power free ω-word over Σ is also given.