The anick complex and the hochschild cohomology of the manturov (2,3)-group

The Manturov $(2,3)$-group $G_3^2$ is the group generated by three elements $a$, $b$, and $c$ with defining relations $a^2=b^2=c^2=(abc)^2=1$. We explicitly calculate the Anick chain complex for $G_3^2$ by algebraic discrete Morse theory and evaluate the Hochschild cohomology groups of the group algebra $\Bbbk G_3^2$ with coefficients in all 1-dimensional bimodules over a field $\Bbbk $ of characteristic zero.