The Anick Complex and the Hochschild Cohomology of the Manturov (2,3)-Group

Abstract

The Manturov (2, 3)-group G32 is the group generated by three elements a, b, and c with defining relations a2 = b2 = c2 = (abc)2 = 1. We explicitly calculate the Anick chain complex for G32 by algebraic discrete Morse theory and evaluate the Hochschild cohomology groups of the group algebra $$\mathbb{k}G_3^2$$ with coefficients in all 1-dimensional bimodules over a field $$\mathbb{k}$$ of characteristic zero.