The cohomology ring of an HNN extension of combinatorially aspherical groups (Q1175212)

Let \(G\) be a combinatorially aspherical group (CA) and \(H\) an HNN extension with base group \(G\). It is known [\textit{J. Huebschmann}, J. Pure Appl. Algebra 14, 137-143 (1979; Zbl 0396.20021)] that a CA group \(G\) has period 2 after 2-steps, i.e. the functors \(H^ i(G,-)\) and \(H^{i+2}(G,-)\) are naturally equivalent for \(i>2\). Here it is shown that if either the associated subgroup of \(H\) or the base group is finitely related then \(H\) has period \(2\ell\) after 3-steps, and that if \(\ell>1\) then the periodicity isomorphisms are induced by cup product with an element of \(H^{2\ell}(H,\mathbb{Z})\). Moreover, if \(H\) is finitely presented and \(R\) a PID then \(H^*(H,R)\) is finitely presented as a ring over \(R\) and a presentation may be found.