Gröbner–Shirshov basis and Hochschild cohomology of the group $\Gamma ^4_5$

In this paper, we construct a Gröbner—Shirshov basis for the group $\Gamma^4_5$ with respect to the tower order on the words. By using this result, we apply the discrete algebraic Morse theory to find explicitly the first two differentials of the Anick resolution for $\Gamma^4_5$, and calculate the first and second Hochschild cohomology groups of the group algebra of $\Gamma^4_5$ with coefficients in the trivial $1$-dimensional bimodule over a field $\mathbb{k}$ of characteristic zero.