Geodesic
Gr\"obner--Shirshov basis and Hochschild cohomology of the group $\Gamma ^4_5$
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 1, pp. 211-236

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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.
Keywords: Gröbner—Shirshov basis, Anick resolution, Hochschild cohomology. 
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     title = {Gr\"obner--Shirshov basis and {Hochschild} cohomology of the group $\Gamma ^4_5$},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
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Hassan Alhussein. Gr\"obner--Shirshov basis and Hochschild cohomology of the group $\Gamma ^4_5$. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 19 (2022) no. 1, pp. 211-236. http://geodesic.mathdoc.fr/item/SEMR_2022_19_1_a7/