Past Algebra Seminar

 Recently several conjectures about l2-invariants of
CW-complexes have been disproved. At the heart of the counterexamples
is a method of computing the spectral measure of an element of the
complex group ring. We show that the same method can be used to
compute the finite field analog of the l2-Betti numbers, the homology
gradient. As an application we point out that (i) the homology
gradient over any field of characteristic different than 2 can be an
irrational number, and (ii) there exists a CW-complex whose homology
gradients over different fields have infinitely many different values.
 

String algebras are tame - their finite-dimensional representations have been classified - and the Auslander-Reiten quiver of such an algebra shows some of the morphisms between them.  But not all.  To see the morphisms which pass between components of the Auslander-Reiten quiver, and so obtain a more complete picture of the category of representations, we should look at certain infinite-dimensional representations and use ideas and techniques from the model theory of modules.

This is joint work with Rosie Laking and Gena Puninski:
G. Puninski and M. Prest,  Ringel's conjecture for domestic string algebras, arXiv:1407.7470;
R. Laking, M. Prest and G. Puninski, Krull-Gabriel dimension of domestic string algebras, in preparation.

The Springer Correspondence relates irreducible representations of the Weyl group of a semisimple complex Lie algebra to the geometry of the cone of nilpotent elements of the Lie algebra. The zeroth Poisson homology of a variety is the quotient of all functions by those spanned by Poisson brackets of functions. I will explain a conjecture with Proudfoot, based on a conjecture of Lusztig, that assigns a grading to the irreducible representations of the Weyl group via the Poisson homology of the nilpotent cone. This conjecture is a kind of symplectic duality between this nilpotent cone and that of the Langlands dual. An analogous statement for hypertoric varieties is a theorem, which relates a hypertoric variety with its Gale dual, and assigns a second grading to its de Rham cohomology, which turns out to coincide with a different grading of Denham using the combinatorial Laplacian.

Let G be a finite group, p a prime and S a Sylow p-subgroup. The group G

is called p-nilpotent if S has a normal complement N in G, that is, G is

the semidirect product between S and N. The notion of p-nilpotency plays

an important role in finite group theory. For instance, Thompson's

criterion for p-nilpotency leads to the important structural result that

finite groups with fixed-point-free automorphisms are nilpotent.

By a classical result of Tate one can detect p-nilpotency using mod p

cohomology in dimension 1: the group G is p-nilpotent if and only if the

restriction map in cohomology from G to S is an isomorphism in dimension

1. In this talk we will discuss cohomological criteria for p-nilpotency by

Tate, and Atiyah/Quillen (using high-dimensional cohomology) from the

1960s and 1970s. Finally, we will discuss how one can extend Tate's

result to study p-solvable and more general finite groups.

Gromov and Piatetski-Shapiro proved the existence of finite volume non-arithmetic hyperbolic manifolds of any given dimension. In dimension four and higher, we show that there are about $v^v$ such manifolds of volume at most $v$, considered up to commensurability. Since the number of arithmetic ones tends to be polynomial, almost all hyperbolic manifolds are non-arithmetic in an appropriate sense. Moreover, by restricting attention to non-compact manifolds, our result implies the same growth type for the number of quasi- isometry classes of lattices in $SO(n,1)$. Our method involves a geometric graph-of-spaces construction that relies on arithmetic properties of certain quadratic forms.

A joint work with Arie Levit.

In 1878, Jordan showed that if $G$ is a finite group of complex $n \times n$ matrices, then $G$ has a normal subgroup whose index in $G$ is bounded by a function of $n$ alone. He showed only existence, and early actual bounds on this index were far from optimal. In 1985, Weisfeiler used the classification of finite simple groups to obtain far better bounds, but his work remained incomplete when he disappeared. About eight years ago, I obtained the optimal bounds, and this work has now been extended to subgroups of all (complex) classical groups. I will discuss this topic at a “colloquium” level – i.e., only a rudimentary knowledge of finite group theory will be assumed.

The common convention when dealing with hyperbolic groups is that such groups are finitely

generated and equipped with the word length metric relative to a finite symmetric generating

subset. Gromov's original work on hyperbolicity already contained ideas that extend beyond the

finitely generated setting. We study the class of locally compact hyperbolic groups and elaborate

on the similarities and differences between the discrete and non-discrete setting.

Let $A$ and $B$ be boundaries of CAT(0) spaces. A function $f:A \to B$ is called a {\em boundary isomorphism} if $f$ is a homeomorphism in the visual topology and

$f$ is an isometry in the Tits metric. A compact metrizable space $Y$ is said to be {\em Tits rigid}, if for any two CAT(0) group boundaries $Z_1$ and $Z_2$ homeomorphic to $Y$, $Z_1$ is boundary isomorphic to $Z_2$.

We prove that the join of two Cantor sets and its suspension are Tits rigid.

A finitely generated group has the Haagerup property if it admits a proper isometric action on a Hilbert space. It was a long open question whether Haagerup property is a quasi-isometry invariant. The negative answer was recently given by Mathieu Carette, who constructed two quasi-isometric generalized Baumslag-Solitar groups, one with the Haagerup property, the other not. Elaborating on these examples, we proved (jointly with S. Arnt and T. Pillon) that the equivariant Hilbert compression is not a quasi-isometry invariant. The talk will be devoted to describing Carette's examples.

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.