This project is dedicated to studying a geometric invariant called the Bogomolov multiplier. The main objectives of the proposed project are threefold.
First of all, we wish to understand how the structure of the Bogomolov multiplier depends on the structure of the underlying group. To this end, we set to inspect the behavior of the Bogomolov multiplier with respect to another group theoretical invariant, the coclass. In turn, this will require thoroughly developing a theory of Bogomolov multipliers associated to profinite groups. A particular instance of these are $p$-adic Lie groups. We aim to enrich our understanding of their Bogomolov multipliers by translating the study to their associated Lie algebras.
Secondly, we are interested in applications of our knowledge about the Bogomolov multiplier. Our focus here will be to strengthen the visible connections between the Bogomolov multiplier and automorphism groups. Kang and Kunyavskii recently noted a link between the Bogomolov multiplier and the Tate-Shafarevich set. This relation is expressible in terms of outer automorphisms of a given group. We aim to prove the implication that groups possessing special outer automorphisms must have nontrivial Bogomolov multipliers. Further evidence of this interplay between automorphisms and Bogomolov multipliers can be seen in the category of representations of a given group as shown by Davydov, and we intend to look into these more abstract aspects as well.
Lastly, we propose to explore some extensions of the Bogomolov multiplier to higher dimensions. Our intentions here are to find algebraic descriptions of higher dimensional unramified cohomology groups and of Ekedahl invariants akin to the combinatorial description of the Bogomolov multiplier. Peyre has shown that this can be achieved for unramified cohomology groups of degree three for a special class of groups. We see a possible extension of these results in terms of higher dimensional combinatorial objects.
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