## 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.

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.

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.

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.