Positively \(n\)-Engel groups. (Q2769827)

The aim of this article is to compare group identities with semigroup identities (equations avoiding inverses) particularly for nilpotence and Engel conditions. Let \(x,y,z_0,z_1,\dots\) be a sequence of (variable) elements, and define words \(f_i\), \(g_i\) inductively by \(f_0=x\), \(g_0=y\), \(f_{n+1}=f_nz_ng_n\), \(g_{n+1}=g_nz_nf_n\). Then \(f_n=g_n\) is the \(n\)-th Mal'cev law. A group is called positively \(n\)-Engel if \(f_n=g_n\) is true for the special cases that all \(z_i\) are \(1\) and that all \(z_i\) are \(xy\). -- The main result is the following statement: If a finitely generated residually finite group \(G\) is positively \(n\)-Engel for some \(n\), then \(G\) is nilpotent of class depending on \(n\) and the number of generators only. Corollary: There are functions \(s(n)\), \(t(n)\) such that positively \(n\)-Engel groups are \(s(n)\)-Engel and \(n\)-Engel groups are positively \(t(n)\)-Engel as long as the groups considered are residually finite.