Past Algebra Seminar

A virtual endomorphism of a group $G$ is a homomorphism $f : H \rightarrow G$ where $H$

is a subgroup of $G$ of fi…nite index $m$: A recursive construction using $f$ produces a

so called state-closed (or, self-similar in dynamical terms) representation of $G$ on

a 1-rooted regular $m$-ary tree. The kernel of this representation is the $f$-core $(H)$;

i.e., the maximal subgroup $K$ of $H$ which is both normal in G and is f-invariant.

Examples of state-closed groups are the Grigorchuk 2-group and the Gupta-

Sidki $p$-groups in their natural representations on rooted trees. The affine group

$Z^n \rtimes GL(n;Z)$ as well as the free group $F_3$ in three generators admit state-closed

representations. Yet another example is the free nilpotent group $G = F (c; d)$ of

class c, freely generated by $x_i (1\leq i \leq d)$: let $H = \langle x_i^n | \

(1 \leq i \leq d) \rangle$ where $n$ is a

fi…xed integer greater than 1 and $f$ the extension of the map $x^n_i

\rightarrow x_i$ $(1 \leq i \leq d)$.

We will discuss state-closed representations of general abelian groups and of

…nitely generated torsion-free nilpotent groups.

The maximal subgroups of the exceptional groups of Lie type

have been studied for many years, and have many applications, for

example in permutation group theory and in generation of finite

groups. In this talk I will survey what is currently known about the

maximal subgroups of exceptional groups, and our recent work on this

topic. We explore the connection with extending morphisms from finite

groups to algebraic groups.

We define the notion of orbit decidability in a general context, and descend to the case of groups to recognise it into several classical algorithmic problems. Then we shall go into the realm of free groups and shall analise this notion there, where it is related to the Whitehead problem (with many variations). After this, we shall enter the negative side finding interesting subgroups which are orbit undecidable. Finally, we shall prove a theorem connecting orbit decidability with the conjugacy problem for extensions of groups, and will derive several (positive and negative) applications to the conjugacy problem for groups.

In this talk, I will explain part of the programme of Gorenstein, Lyons

and Solomon (GLS) to provide a new proof of the CFSG. I will focus on

the difference between the initial notion of groups of characteristic

$2$-type (groups like Lie type groups of characteristic $2$) and the GLS

notion of groups of even type. I will then discuss work in progress

with Capdeboscq to study groups of even type and small $2$-local odd

rank. As a byproduct of the discussion, a picture of the structure of a

finite simple group of even type will emerge.

For a finitely presented group, the Word Problem asks for an algorithm

which declares whether or not words on the generators represent the

identity. The Dehn function is the time-complexity of a direct attack

on the Word Problem by applying the defining relations.

A "hydra phenomenon" gives rise to novel groups with extremely fast

growing (Ackermannian) Dehn functions. I will explain why,

nevertheless, there are efficient (polynomial time) solutions to the

Word Problems of these groups. The main innovation is a means of

computing efficiently with compressed forms of enormous integers.

This is joint work with Will Dison and Eduard Einstein.

We show that for every $n\ge 2$ there exists a torsion-free one-ended word-hyperbolic group $G$ of rank $n$ admitting generating $n$-tuples $(a_1,\ldots ,a_n)$ and $(b_1,\ldots ,b_n)$ such that the $(2n-1)$-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{1,\ldots ,1}_{n-1 \text{ times}} )$$ are not Nielsen-equivalent in $G$. The group $G$ is produced via a probabilistic construction (joint work with Ilya Kapovich).

One of the most classical questions of modern algebra is whether the group algebra of a torsion-free group can be embedded into a skew field. I will give a short survey about embeddability of group algebras into skew fields, matrix rings and, in general, continuous rings.

Given a residually finite group, we analyse a growth function measuring the minimal index of a normal subgroup in a group which does not contain a given element. This growth (called residual finiteness growth) attempts to measure how ``efficient'' of a group is at being residually finite. We review known results about this growth, such as the existence of a Gromov-like theorem in a particular case, and explain how it naturally leads to the study of a second related growth (called intersection growth). Intersection growth measures asymptotic behaviour of the index of the intersection of all subgroups of a group that have index at most n. In this talk I will introduce these growths and give an overview of some cases and properties.

This is joint work with Ian Biringer, Khalid Bou-Rabee and Martin Kassabov.

Let $\Gamma$ be a group and let $r_n(\Gamma)$ denote the

number of isomorphism classes of irreducible $n$-dimensional complex

characters of $\Gamma$. Representation growth is the study of the

behaviour of the numbers $r_n(\Gamma)$. I will give a brief overview of

representation growth.

We say $\Gamma$ has polynomial representation growth if $r_n(\Gamma)$ is

bounded by a polynomial in $n$. I will discuss a question posed by

Brent Everitt: can a group with polynomial representation growth have

the alternating group $A_n$ as a quotient for infinitely many $n$?

I will review several known problems on the automorphism group of finite $p$-groups and present a sketch of the proof of the the following result obtained jointly with Jon Gonz\'alez-S\'anchez:

For each prime $p$ we construct a family $\{G_i\}$ of finite $p$-groups such that $|Aut (G_i)|/|G_i|$ goes to $0$, as $i$ goes to infinity. This disproves a well-known conjecture that $|G|$ divides $|Aut(G)|$ for every non-abelian finite $p$-group $G$.

Thoughts on the Burnside problem

Let F be a field. A symplectic alternating algebra over F

consists of a symplectic vector space V over F with a non-degenerate

alternating form that is also equipped with a binary alternating

product · such that the law (u·v, w)=(v·w, u) holds. These algebraic

structures have arisen from the study of 2-Engel groups but seem also

to be of interest in their own right with many beautiful properties.

We will give an overview with a focus on some recent work on the

structure of nilpotent symplectic alternating algebras.

Affinoid enveloping algebras arise as certain p-adic completions of ordinary enveloping algebras, and are closely related to Iwasawa algebras. I will explain how to use Beilinson-Bernstein localisation to compute their (non-commutative) Krull dimension. This is recent joint work with Ian Grojnowski.

Let g be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, b a Borel subalgebra of g, h the Cartan sublagebra contained in b and N the unipotent subgroup corresponding to the nilradical n of b. Extremal projection operators are projection operators onto the subspaces of n-invariants in certain g-modules the action of n on which is locally nilpotent. Zhelobenko also introduced a family of operators which are analogues to extremal projection operators. These operators are called now Zhelobenko operators.
I shall show that the explicit formula for the extremal projection operator for g obtained by Asherova, Smirnov and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence (N, h) -> b given by the restriction of the adjoint action. Simple geometric proofs of  formulas for the ``classical'' counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.

We will first recall the known notion of commensurating actions

and its link to actions on CAT(0) cube complexes. We define a

group to have Property FW if every isometric action on a CAT(0)

cube complex has a fixed point. We conjecture that every

irreducible lattice in a semisimple Lie group of higher rank has

Property FW, and will give some instances beyond the trivial

case of Kazhdan groups.

The spectral presheaf of a nonabelian von Neumann algebra or C*-algebra

was introduced as a generalised phase space for a quantum system in the

so-called topos approach to quantum theory. Here, it will be shown that

the spectral presheaf has many features of a spectrum of a

noncommutative operator algebra (and that it can be defined for other

classes of algebras as well). The main idea is that the spectrum of a

nonabelian algebra may not be a set, but a presheaf or sheaf over the

base category of abelian subalgebras. In general, the spectral presheaf

has no points, i.e., no global sections. I will show that there is a

contravariant functor from unital C*-algebras to their spectral

presheaves, and that a C*-algebra is determined up to Jordan

*-isomorphisms by its spectral presheaf in many cases. Moreover, time

evolution of a quantum system can be described in terms of flows on the

spectral presheaf, and commutators show up in a natural way. I will

indicate how combining the Jordan and Lie algebra structures may lead to

a full reconstruction of nonabelian C*- or von Neumann algebra from its

spectral presheaf.

The spectral presheaf of a nonabelian von Neumann algebra or C*-algebra was introduced as a generalised phase space for a quantum system in the so-called topos approach to quantum theory. Here, it will be shown that the spectral presheaf has many features of a spectrum of a noncommutative operator algebra (and that it can be defined for other classes of algebras as well). The main idea is that the spectrum of a nonabelian algebra may not be a set, but a presheaf or sheaf over the base category of abelian subalgebras. In general, the spectral presheaf has no points, i.e., no global sections. I will show that there is a contravariant functor from unital C*-algebras to their spectral presheaves, and that a C*-algebra is determined up to Jordan *-isomorphisms by its spectral presheaf in many cases. Moreover, time evolution of a quantum system can be described in terms of flows on the spectral presheaf, and commutators show up in a natural way. I will indicate how combining the Jordan and Lie algebra structures may lead to a full reconstruction of nonabelian C*- or von Neumann algebra from its spectral presheaf.

In my talk I shall give a small survey on some algorithmic properties of amalgamated products of finite rank 
free groups. In particular, I'm going to concentrate on Membership Problem for this groups. Apart from being algorithmically interesting, amalgams of free groups admit a lot of interpretations. I shall show how to 
characterize this construction from the point of view of geometry and linguistic.