Past Algebra Seminar

A polar space $\Pi$ is a geometry whose elements are the totally isotropic subspaces of a vector space $V$ with respect to either an alternating, Hermitian, or quadratic form. We may form a new geometry $\Gamma$ by removing all elements contained in either a hyperplane $F$ of $\Pi$, or a hyperplane $H$ of the dual $\Pi^*$. This is a \emph{biaffine polar space}.

We will discuss two specific examples, one with automorphism group $q^6:SU_3(q)$ and the other $G_2(q)$. By considering the stabilisers of a maximal flag, we get an amalgam, or "glueing", of groups for each example. However, the two examples have "similar" amalgams - this leads to a group recognition result for their automorphism groups.

I shall discuss recent work in which bounds are obtained, generalising/specialising earlier work for general linear groups

Direct products of finitely generated free groups have a surprisingly rich subgroup structure. We will talk about how the finiteness properties of a subgroup of a direct product relate to the way it is embedded in the ambient product. Central to this connection is a conjecture on finiteness properties of fibre products, which we will present along with different approaches towards solving it.

In 1968, Dixmier posed six problems for the algebra of polynomial

  differential operators, i.e. the Weyl algebra. In 1975, Joseph

solved the third and sixth problems and, in 2005, I solved the

  fifth problem and gave a positive solution to the fourth problem

  but only for homogeneous differential operators. The remaining three problems are still open. The first problem/conjecture of Dixmier (which is equivalent to the Jacobian Conjecture as was shown in 2005-07 by Tsuchimito, Belov and Kontsevich) claims that the Weyl algebra `behaves'

like a finite field. The first problem/conjecture of

  Dixmier:   is it true that an algebra endomorphism of the Weyl

  algebra an automorphism? In 2010, I proved that this question has

  an affirmative answer for the algebra of polynomial

  integro-differential operators. In my talk, I will explain the main

  ideas, the structure of the proof and recent progress on the first problem/conjecture of Dixmier.

In 1939, Wilhelm Magnus gave a characterization of free groups in terms of their rank and nilpotent quotients. Our goal in this talk is to present results giving both positive and negative answers to the following question: does a similar characterization hold within the class of finite-extensions of finitely generated free groups? This talk covers joint work with Brandon Seward.

Word maps on groups were studied extensively in the past few years, in connection to various conjectures on profinite groups, finite groups, finite simple groups, etc. I will provide background, as well as very recent works (joint with Larsen, Larsen-Tiep,

Liebeck-O'Brien-Tiep) on word maps with relations to representations (e.g. Gowers' method and character ratios), geometry and probability.

Recent applications, e.g. to subgroup growth and representation varieties, will also be described.

I will conclude with a list of problems and conjectures which are still very much open.  The talk should be accessible to a wide audience.

Following the works of Sela and Kharlampovich-Myasnikov on the Tarski problem, we are interested in the first-order logic of free (and more generally hyperbolic) groups. It turns out that techniques from geometric group theory can be used to answer many questions coming from model theory on these groups. We showed with Sklinos that free groups of finite rank are homogeneous, namely that two tuples of elements which have the same first-order properties are in the same orbit under the action of the automorphism group. We also show that this is not the case for most surface groups.

We analyze the question of the minimal index of a normal subgroup in a free group which does not contain a given element. Recent work by BouRabee-McReynolds and Rivin give estimates for the index. By using results on the length of shortest identities in finite simple groups we recover and improve polynomial upper and lower bounds for the order of the quotient. The bounds can be improved further if we assume that the element lies in the lower central series.

I will prove that generating sets of surface groups are either reducible or Nielsen equivalent to standard generating sets, improving upon a theorem of Zieschang. Equivalently, Aut(F_n) acts transitively on Epi(F_n,S) when S is a surface group.

Abstract:

this is joint work with Eli Aljadeff.

Let G be a group, H a finite index subgroup. Moore's conjecture says that under a certain condition on G and H (which we call the Moore's condition), a G-module M which is projective over H is projective over G. In other words- if we know that a module is ``almost projective'', then it is projective. In this talk we will survey cases in which the conjecture is known to be true. This includes the case in which the group G is finite and the case in which the group G has finite cohomological dimension.

As a generalization of these two cases, we shall present Kropholler's hierarchy LHF, and discuss the conjecture for groups in this hierarchy. In the case of finite groups and in the case of finite cohomological dimension groups, the conjecture is proved by the same finiteness argument. This argument is straightforward in the finite cohomological dimension case, and is a result of a theorem of Serre in case the group is finite. We will show that inside Kropholler's hierarchy the conjecture holds even though this finiteness condition might fail to hold.

We will also discuss some other cases in which the conjecture is known to be true (e.g. Thompson's group F).