Word maps with constants on symmetric groups

We study word maps with constants on symmetric groups. Even though there are non‐trivial mixed identities of bounded length that are valid for all symmetric groups, we show that no such identities can hold in the limit in a metric sense. Moreover, we prove that word maps with constants and non‐trivial content, that are short enough, have an image of positive diameter, measured in the normalized Hamming metric, which is bounded from below in terms of the word length. Finally, we also show that every self‐map G→G$G\rightarrow G$ on a finite non‐abelian simple group is actually a word map with constants from G.