Groups with bounded verbal conjugacy classes
Sergio Brazil, Alexei N. Krasilnikov, Pavel Shumyatsky · Journal of Group Theory · 2006
Let F be a free group and let w ∈ F . For a group G , let G w denote the set of all w -values in G and w ( G ) the verbal subgroup of G corresponding to w . A word w is called boundedly concise if, for each group G such that | G w | ≤ m , we have | w ( G )| ≤ c for some integer c = c ( m ) depending only on m . The main theorem of the paper says that if w is a boundedly concise word and G is a group such that | x G w | ≤ m for all x ∈ G then | x w ( G ) | ≤ d for all x ∈ G and some integer d = d ( m , w ) depending only on m and w .