Groups that satisfy ${ [x,y]^2 \approx 1 }$ Lately, I have been constructing finite involution monoids that generate varieties with $2^{\aleph_0}$ subvarieties. One construction requires groups that violate the identity ${ [x,y]^2 \approx 1 }$, where ${ [x,y] = x^{-1} y^{-1} xy }$. 
Is there a name for groups satisfying the identity ${ [x,y]^2 \approx 1 }$? Has there been any work done on these groups?
 A: Not every group satisfying the law $[x,y]^2=1$ is metabelian. But by Theorem 4 of McDonald, I. D. "On certain varieties of groups" 
Math. Z. 76 1961 270–282. every group satisfying this law has second derived subgroup in the center so it is center-by-metabelian, and the derived subgroup of exponent 4. 
Note. I had a wrong answer before and did not notice Farrokhi's answer when I made corrections - one hour after his answer. Farrokhi's answer is more complete. 
A: Here are some additional details for the answer of M. Farrokhi.  In the paper " On a conjecture of Hanna Neumann", B.H.Neumann constructs a certain group $G$.  There are generators $a_1,\dotsc,a_4$, and additional elements defined in terms of these as follows:
\begin{align*}
 b_{12} &= [a_1,a_2] = [a_2,a_1] \\
 b_{13} &= [a_1,a_3] = [a_3,a_1] \\
 b_{14} &= [a_1,a_4] = [a_4,a_1] \\
 b_{23} &= [a_2,a_3] = [a_3,a_2] \\
 b_{24} &= [a_2,a_4] = [a_4,a_2] \\
 b_{34} &= [a_3,a_4] = [a_4,a_3] \\
 c_1 &= [a_2,b_{34}] = [a_4,b_{23}] \\
 c_2 &= [a_3,b_{14}] = [a_4,b_{13}] \\
 c_3 &= [a_4,b_{12}] = [a_1,b_{24}] \\
 d &= [a_1,[a_2,[a_3,a_4]] = [a_2,[a_3,[a_4,a_1]] = [a_3,[a_4,[a_1,a_2]]
\end{align*}
There are some relations implicit in the above equations.  There are also additional relations as follows:


*

*$a_i^2=b_{jk}^2=c_l^2=d^2=1$

*All commutators $[a_i,b_{jk}]$ that have not already been listed, are trivial.

*$[a_i,c_j]=1$ whenever $i\neq j$, and $[a_i,d]=1$


We can define a map $\phi\colon\{0,1\}^{14}\to G$ by 
$$ \phi(u) = a_1^{u_1}\dotsb a_4^{u_4} b_{12}^{u_5} \dotsb b_{34}^{u_{10}} 
    c_1^{u_{11}}c_2^{u_{12}}c_3^{u_{13}}d^{u_{14}}
$$
One can check that this is bijective, and one can write formulae for the permutations of $\{0,1\}^{14}$ corresponding to right multiplication by the elements $a_i$, $b_{jk}$, $c_l$ and $d$.  In particular, this proves that $|G|=2^{14}$.  One can also check that 
$$ [b_{12},b_{34}] = [b_{13},b_{24}] = [b_{14},b_{23}] = d \neq 1, $$
so $G$ is not metabelian.
In the paper "On certain varieties of groups", Macdonald states that it is easy to verify that the above group has $[x,y]^2=1$ for all $x,y\in G$.  I don't see how to prove this myself.  However, I have checked it by computer for 10000 randomly chosen pairs $(x,y)$, so it must be true.
A: Here are some preliminary results.  Let $F$ be freely generated by $x$ and $y$, let $N\leq F$ be generated by all squares of commutators, and put $G=F/N$.  We are interested in the structure of $G$ and $G'$.  For $i,j\in\mathbb{Z}$ put $z_{ij}=[x^i,y^j]$ (which might be interepreted as an element of $F$ or $G$).  I believe that $F'$ is generated by the elements $z_{ij}$ subject only to $z_{0j}=z_{i0}=1$, so $G'$ is also generated by the $z_{ij}$.  In $G$ we have $z_{ij}^2=1$, but there are extra relations as well.  
First, as $z_{ij}^2=z_{kl}^2=1$ we see that $[z_{ij},z_{kl}]=(z_{ij}z_{kl})^2$, and the square of this must be trivial, so $(z_{ij}z_{kl})^4=1$.  If we could improve this to $(z_{ij}z_{kl})^2=1$ then we would see that $G'$ is abelian.
In $F$ one can check that $z_{ip}z_{jp}^{-1}=[x^{i-j},x^jy^p]$, so in $G$ we have $(z_{ip}z_{jp}^{-1})^2=1$ as well as $z_{ip}^2=z_{jp}^2=1$.  From this it follows easily that $z_{ip}$ commutes with $z_{jp}$.  Similarly, we see that $z_{mi}$ commutes with $z_{mj}$.  
Along the same lines, one can check that
$$ [x^iy^j,x^ky^l] = z_{ij}\;z_{i+k,j}^{-1}\;z_{i+k,l}\;z_{k,l}^{-1} $$
so the square of the right hand side is the identity.  Adjacent terms on the right hand side commute in $G$ by the two rules that we have already established.  
This feels like it is getting close, but I am not sure what to do next.
A: This variety of groups has indeed been considered in the literature. It is known that the following conditions hold for every group $G$ satisfying the identity $[x,y]^2=1$:


*

*$[[x,y_1,\ldots,y_m],[x,z_1,\ldots,z_n]]=1$ for all
$x,y_1,\ldots,y_m,z_1,\ldots,z_n\in G$ (see [1]).

*$[[x_1,x_2],[x_3,x_4]]=[[x_{\pi(1)},x_{\pi(2)}],[x_{\pi(3)},x_{\pi(4)}]]$ for all $x_1,\ldots,x_4\in G$ and $\pi\in S_4$ (see [2]).

*$[[x,y],[z,w]]=[x,y,z,w][x,y,w,z]$ (see [1]).

*$[\gamma_2(G),\gamma_3(G)]=1$ (see [2]).

*$[G'',G]=1$ (see [2]).

*$G'^4=1$ (see [2]).

*The group $G$ need not be metabelian (see [3]). 

*If $G$ is finite, then $G=P\rtimes H$, where $P$ is a normal Sylow $2$-subgroup of $G$, $H$ is abelian of odd order, and $[P,H]$ is an elementary abelian $2$-group (see [1]).


References:


*

*M. Farrokhi D. G., On groups satisfying a symmetric Engel word, Ric. Mat. 65 (2016), 15–20.

*I. D. Macdonald, On certain varieties of groups, Math. Z. 76, (1961) 270–282.

*B. H. Neumann, On a conjecture of Hanna Neumann, Proc. Glasgow Math. Assoc. 3 (1956), 13–17.
