We prove that in the Polish group of isometries of the Baire space the collection of n-th powers is non-Borel. We also prove that in the Polish space of trees on ℕ the collection of trees that have an automorphism under which every node has order exactly n is non-Borel.
"The complexity of squares in the group of isometries of the Baire space." J. Symbolic Logic 77 (1) 329 - 336, March 2012. https://doi.org/10.2178/jsl/1327068706