The set of squares in the group of autohomeomorphisms of the circle is complete analytic, and hence analytic but not Borel.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm195-2-2, author = {Paul Gartside and Bojana Peji\'c}, title = {The complexity of the set of squares in the homeomorphism group of the circle}, journal = {Fundamenta Mathematicae}, volume = {193}, year = {2007}, pages = {125-134}, zbl = {1165.54015}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm195-2-2} }
Paul Gartside; Bojana Pejić. The complexity of the set of squares in the homeomorphism group of the circle. Fundamenta Mathematicae, Tome 193 (2007) pp. 125-134. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm195-2-2/