On subsemigroup lattices of aperiodic groups (Q1318956)

The paper concerns subsemigroups of aperiodic groups. The main results are the following two theorems: Theorem 1. There are continuously many non-isomorphic 2-generated torsion-free groups each group \(G\) of which has the properties that every maximal subsemigroup of \(G\) is a cyclic subgroup of \(G\) and different maximal subgroups of \(G\) intersect trivially. Theorem 2. Given a prime \(n \gg 1\) (e.g. \(n > 10^{80}\)) there are continuously many non-isomorphic 2-generated aperiodic groups each group \(G\) of which has the following two properties: (a) the semigroup law \(x^ n(x^ n y^ n)^ n = (x^ n y^ n)^ n x^ n\) holds in \(G\); (b) every maximal subsemigroup of \(G\) is a cyclic subgroup of \(G\) either of infinite order or of order \(n\) and different maximal subgroups of \(G\) intersect trivially.