Forthcoming events in this series
D-modules and arithmetic: a theory of the b-function in positive characteristic.
Abstract
We exhibit a construction in noncommutative nonnoetherian algebra that should be understood as a positive characteristic analogue of the Bernstein-Sato polynomial or b-function. Recall that the b-function is a polynomial in one variable attached to an analytic function f. It is well-known to be related to the singularities of f and is useful in continuing a certain type of zeta functions, associated with f. We will briefly recall the complex theory and then emphasize the arithmetic aspects of our construction.
Spin projective representations of Weyl groups, Green polynomials, and nilpotent orbits
Abstract
The classification of irreducible representations of pin double covers of Weyl groups was initiated by Schur (1911) for the symmetric group and was completed for the other groups by A. Morris, Read and others about 40 years ago. Recently, a new relation between these projective representations, graded Springer representations, and the geometry of the nilpotent cone has emerged. I will explain these connections and the relation with a Dirac operator for (extended) graded affine Hecke algebras. The talk is partly based on joint work with Xuhua He.
Regular maps and simple groups
Abstract
A regular map is a highly symmetric embedding of a finite graph into a closed surface. I will describe a programme to study such embeddings for a rather large class of graphs: namely, the class of orbital graphs of finite simple groups.
Branch groups: groups that look like trees
Abstract
Groups which act on rooted trees, and branch groups in particular, have provided examples of groups with exotic properties for the last three decades. This and their links to other areas of mathematics such as dynamical systems has made them the object of intense research.
One of their more useful properties is that of having a "tree-like" subgroup structure, in several senses.
I shall explain what this means in the talk and give some applications.
On universal right angled Artin groups
Abstract
the only permitted defining relators are commutators of the generators. These groups and their subgroups play an important role in Geometric Group Theory, especially in view of the recent groundbreaking results of Haglund, Wise, Agol, and others, showing that many groups possess finite index subgroups that embed into RAAGs.
In their recent work on limit groups over right angled Artin groups, Casals-Ruiz and Kazachkov asked whether for every natural number n there exists a single "universal" RAAG, A_n, containing all n-generated subgroups of RAAGs. Motivated by this question, I will discuss several results showing that "universal" (in various contexts) RAAGs generally do not exist. I will also mention some positive results about universal groups for finitely presented n-generated subgroups of direct products of free and limit groups.
Commuting probabilities of finite groups
Abstract
The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Not all rationals between 0 and 1 occur as commuting probabilities. In fact Keith Joseph conjectured in 1977 that all limit points of the set of commuting probabilities are rational, and moreover that these limit points can only be approached from above. In this talk we'll discuss a structure theorem for commuting probabilities which roughly asserts that commuting probabilities are nearly Egyptian fractions of bounded complexity. Joseph's conjectures are corollaries.
Nielsen realisation for right-angled Artin groups
Abstract
We will introduce both the class of right-angled Artin groups (RAAG) and
the Nielsen realisation problem. Then we will discuss some recent progress
towards solving the problem.
17:00
On computing homology gradients over finite fields
Abstract
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.
Ziegler spectra of domestic string algebras
Abstract
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.
Sandpile groups of Eulerian digraphs and an explicit presentation for the group of units in F_p[Z_n]
The Springer Correspondence and Poisson homology
Abstract
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.
A theorem of Tate and p-solvability
Abstract
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.
Counting commensurability classes of hyperbolic manifolds
Abstract
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.
Finite subgroups of the classical groups
Abstract
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.
Locally compact hyperbolic groups
Abstract
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.
The subgroup structure of automorphism groups of a partially commutative groups
Tits rigidity of CAT(0) group boundaries
Abstract
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.
The Haagerup property is not a quasi-isometry invariant (after M. Carette)
Abstract
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.
Virtual Endomorphisms of Groups
Abstract
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.
Maximal subgroups of exceptional groups of Lie type and morphisms of algebraic groups
Abstract
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.