What are the "freest" finitely generated $2-Engel$ groups?

Question

Let $G$ be a group generated by $x_1,\dots,x_n$. Suppose that we also require that $G$ to be a $2-$Engel group (that is, $x$ commutes with $g^{-1}xg$ for all $x,g\in G$) but there are no other relations. Is $G$ isomorphic to some familiar group?

Answer 1

There's a notion of group variety (a class of groups closed under taking factors, subgroups and direct products). Birkhoff theorem says that it's equivalent to classes of groups satisfying a fixed system of identity relations. Any variety has free groups; namely, groups with presentation consisting of defining relations. So the freest 2-Engel group is free 2-Engel group on alphabet X $\langle X \, | \, [a, b, b] = 1 \rangle$, where $a$ and $b$ vary over all words in $X \sqcup X^{-1}$.

Engel groups are pretty special and there's no good way to express free Engel groups in more or less elementary terms. A way to see what is $E(2, n)$ is to write a few identities which hold in it and obtain some canonical factors.

In 2-Engel group $G$

1. $\gamma_4(G) = 1$ (it follows that $G$ is metabelian);
2. $\gamma_3(G)$ is 3-torsion;
3. $[x, y, z]$ is cyclic invariant and antisymmetric in last two variables;
4. $[x^n, y] = [x, y]^n = [x, y^n]$ for any $n$
5. Free 2-step nilpotent group is 2-Engel

Consequently, in $E(2, n)$ free on ordered generators $(x_1, \dots, x_n)$ every element has a unique normal form $$\prod_{1 \leq i \leq n} ^{ordered} x_i^{\theta_i} \prod_{1 \leq j \leq k \leq n}[x_j, x_k]^{\eta_{jk}} \prod_{1 \leq a \leq b \leq c \leq n}[x_a, x_b, x_c]^{\tau_{abc}}; \quad \theta, \eta \in \mathbb Z, \tau \in \mathbb Z/3$$ and we have

• central extension $\Lambda^3(\mathbb Z^n) \otimes \mathbb Z/3 \hookrightarrow E(2, n) \twoheadrightarrow Nil(2, n)$
• extension $(E (2, n))^3 \hookrightarrow E (2, n) \twoheadrightarrow Burn(3, n)$ ($Burn(3, n)$ is Burnside group of exponent 3)

Engel groups of higher class are way more complicated; there's no known normal form for elements in 3-Engel groups; $E(4, 3)$ already has nilpotency class 9 and its 5-torsion is $(\mathbb Z / 5)^{57}$; every free nonabelian 6-Engel group has 7-torsion. It's an open question whether finitely generated n-Engel groups nilpotent or not.