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# Bogomolov Multiplier

## Periodic Reporting for period 1 - BogomolovMultiplier (Bogomolov Multiplier)

Reporting period: 2017-10-01 to 2019-09-30

This project can roughly be situated inside mathematics in a branch called
group theory. This theory is a mathematical foundation of the concept of
symmetry. Its beginnings go back to the study of solutions of polynomial
equations as envisioned by Galois in 1830s. More precisely, Galois showed
how to associate a group to a polynomial, and then use this group to deduce
some properties about the roots of the polynomial. This idea was later made
more abstract and the concept of an abstract group is present all over
modern mathematics.

One of the fundamental unresolved problems in the theory of groups asks
whether or not every finite abstract group arises from a polynomial whose
coefficients are rational numbers. This question therefore asks for an inverse
procedure to the construction of Galois. It is known to have a positive solution
for polynomials with more complicated coefficients than rational numbers
(for example, for rational functions with complex coefficients).

One can try to imitate the positive solution for some other
coefficients to get an answer to the inverse Galois problem over rational
numbers. These ideas led Noether to establish a programme on how to construct
polynomials with a given finite abstract group G associated to them. The
idea is to associate a certain (regular) representation of the abstract group
on a vector space and then quotient by the action of G. One gets an
algebraic variety, given by a set of polynomial equations. Noether conjectured
that there is a way of solving these polynomial equations in terms of simple
rational functions (meaning that the quotient variety would be what is called
rational in algebraic geometry), and this would be enough to imply that her
construction produces many polynomials with Galois group G.

It turned out much later that Noether's problem does not always have a
positive solution. Obstructions were developed to show that the polynomial
equations arising from her constructions can not be solved as she imagined,
and concrete groups G were presented for which these obstructions are
non-trivial. This project dealt with understanding possibly the most basic of
these obstructions, nowadays called the Bogomolov multiplier and denoted by
B_0(G). The aim was to understand various structural aspects of this
obstruction in relation to the abstract group G, use this to produce more
negative examples, and to explore some extensions of this obstruction.
The relevance of this, apart for the original motivation regarding the
inverse Galois problem, is that many diverse interpretations of the
Bogomolov multiplier are known, making this object a meeting-point for
several important areas of mathematics such as geometry, homology,
K-theory, representation theory, mathematical physics.
In WP1, we extended previous results we had about Bogomolov multipliers of
p-groups of maximal class to arbitrary coclass. We investigated the structure
of Bogomolov multipliers as well as Schur multipliers from an asymptotic point
of view. We studied Bogomolov multipliers of p-adic space groups and gave
upper bounds on their structural parameters. We also considered nilpotent
p-adic Lie groups. We showed how to transfer the study of their Bogomolov
multipliers into a combinatorial problem in Lie rings, and we considered the
case of rings of nilpotency class 2 in particular. Here we could derive that
log-generic p-groups have non-trivial Bogomolov multipliers.

In WP2, we considered an open problem posed by Kang and Kunyavskii. We
extended results of Davydov interpreting Bogomolov multipliers and particular
outer automorphisms of G as autoequivalences of the Drinfeld center of G. This
led us to studying lazy cohomology of the group G. We
constructed several sets that generalize this and connected them to the second
homology of G. This led us to a known conjecture stating that whenever an
automorphism acts trivially on all sufficiently high dimensional cohomology
groups, then it must be inner. We focused on the modular case, where we
generalized the notion of Bogomolov multipliers to objects in high dimensional
modular cohomology rings. We were then able to give a conceptual answer to the
above problem relying on works of Symonds and Quillen.

In WP3, we inspected the lowest dimensional generalization of the Bogomolov
multiplier, the third unramified cohomology group. We were working towards
giving explicit examples of groups of smallest possible order with non-trivial
third unramified cohomology and trivial Bogomolov multipliers. We reduced the
problem to finding a certain integral cohomological element in the fourth
dimension cohomology of some p-group G of order p^5. We considered a
generalization of the Evens norm map and we showed that one can detect similar
behaviour as the one we were looking for in some higher dimensional modular
cohomology groups. We were also able to generalize our construction of the
algebraic morphism for understanding generic Bogomolov multipliers to this
setting.

Finally we addressed some other research problems as parts of an independent
work package dealing with word maps. We were able to show that if w is either
the 2-Engel or the metabelian word, then whenever w is not an identity in a
finite group G, the probability that a random tuple of elements in G
satisfies the law w = 1 is absolutely bounded away from 1. We were also
working on words maps of algebraic groups. We considered the simplest case of
these and developed a method of simplifying words values by wiggling one
parameter. We are thus able to effectively check if a single words induces a
surjective map, resolving some open problems in the area.

During the course of this research project, the ER has done several research visits
to other universities, mostly combined with giving seminar talks. He has also given
independent invited talks at several seminars and conferences. He has participated
at several workshops and conferences and has given local talks at the seminar of the
host institution. He has hosted researcher at the local department and organized
several activities, such as reading groups, a video seminar, a PhD course and two
conferences. He has also participated at a mathematical dissemination event for the
general public.
In this project, we addressed old, difficult and fundamental
problems at the intersection between algebra and geometry. As one of the main
results of this project, we have shown that the intuitive expectation that
Noether's approach to the inverse Galois problem morally works is not quite
correct by non-triviality of Bogomolov multipliers of generic p-groups.
Since the central object of interest in the project has many apparitions
throughout mathematics, our contributions have an impact on all of these
areas. We have further found both answers and clear obstructions to them in
other work packages, leading us to expand the project and address as well as
solve problems that were not originally planned. Such is, for example, our
contribution to measuring relative sizes of fibres of word maps, as well as
our exploration of the connection between Bogomolov multipliers and outer
automorphisms. These are all very basic problems, touching various branches of
mathematics and are therefore absolutely relevant in mathematics due to their
abstract nature. Parts of the original results of this project have been
collected into research papers and submitted/published in mathematical
journals.