ИСТИНА

Using results on the topology of polyhedral products, we link distinct concepts of homotopy theory and geometric group theory. On the homotopical side, we describe the Pontryagin algebra (loop homology) of the moment-angle complex ZK. On the group-theoretical side, we describe the structure of the commutator subgroup RC′K of a right-angled Coxeter group RCK, viewed as the fundamental group of the real moment-angle complex RK. For a flag simplicial complex K, we present a minimal generator set for the Pontryagin algebra H∗(ΩZK) and for the commutator subgroup RC′K, and specify a necessary and sufficient combinatorial condition for H∗(ΩZK) and RC′K to be a free or one-relator algebra (group). We also give homological characterisations of these properties. For RC′K, this is given by a condition on the homology group H2(RK), whereas for H∗(ΩZK) this can be stated using the bigrading of the homology of ZK. Parts of this talk are joint works with Jelena Grbic, Marina Ilyasova, George Simmons, Stephen Theriault, Yakov Veryovkin and Jie Wu.