Abstracts

Hadi Bigdely

Combination of groups with hyperbolically embedded subgroups and groups with well-defined relative Dehn functions

Hyperbolic groups play an important role in Geometric Group Theory.  These groups were introduced by Mikhail Gromov.  Various generalizations, such as relatively hyperbolic groups, were late developed.  Hyperbolically embedded subgroups, defined by F. Dahmani, V. Guirardel, and D. Osin, generalize the peripheral structure of relatively hyperbolic groups.  We revisit the definition of these subgroups using the Bowditch graph approach described by E. Martinez Pedroza and F. Rashid.  We then prove a combination theorem for hyperbolically embedded subgroups.  Additionally, after defining groups with a well-defined relative Dehn function, we prove a similar combination theorem for these groups.  This is a joint work with E. Martinez Pedroza.

Alberto Cassella

Hypercubical groups

The hypercubical complex of a finitely generated group (G, S) can be seen as the analog of a flag complex.  When this complex is contractible, the finitely generated group (G, S) is said to be hypercubical.  During this talk we will introduce the definition and some general results about this class of groups, before focusing on some subfamilies of hypercubical groups.  In particular we will show a generalization of right-angled Artin groups and a family of groups defined via an inductive process that begins with the link group of the Borromean rings.

Fabienne Chouraqui

On Garside monoids and groups

Garside groups have been first introduced by P. Dehornoy and L. Paris in 1990.  In many aspects, Garside groups extend braid groups and more generally finite-type Artin groups.  These are torsion-free groups with a word and conjugacy problems solvable, and they are groups of fractions of monoids with a structure of lattice with respect to left and right divisibilities.  It is natural to ask if there are additional properties Garside groups share in common with the intensively investigated braid groups and finite-type Artin groups.  In this talk, I will introduce the Garside groups in general, and a particular class of Garside groups, that arise from certain solutions of the Quantum Yang Baxter equation.  I will describe the connection between these theories arising from different domains of research, present some of the questions raised for the Garside groups and give some partial answers as time permits.

Martina Conte

Conciseness of first-order formulas

A word w is said to be concise if the verbal subgroup w(G) is finite for every group in which w takes only finitely many values.  Concise words were introduced by Philip Hall, who conjectured that every word is concise.  Although this conjecture has been disproved, it is still an open problem to determine whether every word is concise if one restricts to residually finite groups.

The set of word values of any word can be seen as a set defined by a first-order formula in the language of groups.

In this talk I will give an overview on joint work in progress with Moritz Petschick in which we elaborate on this perspective further, generalising the notion of conciseness to certain first-order formulas in the language of groups.

Ged Corob Cook

Hierarchically decomposable tdlc groups

Kropholler's class of hierarchically decomposable groups has many nice properties.  For example, groups G of type FP∞ in this class have a bound on the orders of finite subgroups, and have finite dimensional classifying spaces EℱG with respect to the family ℱ of finite subgroups.  Totally disconnected, locally compact (tdlc) groups introduce some new difficulties, because the category of discrete Z[G]-modules does not have enough projectives, but we can still obtain many similar results to the case of discrete groups.  I will survey what is known, and show that soluble tdlc groups of type FP∞ have finite dimensional classifying spaces with respect to the family of compact open subgroups. 

Alessandro Dioguardi Burgio

Cofiltration of special orthogonal group

The special orthogonal group

SO3 (ℚ)  = {X ∊ M3 (ℚ): XXt = XtX = I3 , det(X) = 1}

has the following property: each of the entries of X ∊ SO3 (ℚ) is a rational number with odd denominator.  This allows us to define the reduction modulo 2 (hence modulo 2k for k = 1, 2, ... ) of the group and p = 2 is the only prime for which this reduction is possible.  We will discuss for which primes p the special orthogonal group of a ternary anisotropic quadratic form over the rational numbers can be reduced modulo p.  Finally, some questions regarding the profiniteness of such a group will be addressed.

Giovanni Falcone

Jacobians of elliptic curves as generalized Jacobians

Let C be an elliptic curve defined over the local field of p-adics, let R be the ring of p-adic integers, and let k = R/pR be the residue field.  Suppose that C has good reduction at p, that is, the image of C under the reduction map is a non singular elliptic curve over the finite field k.  Then it is well-known that the kernel of the reduction map is isomorphic to R.  In this talk we discuss this extension of R by the image of C under the reduction map, comparing it with the extension corresponding to the generalized Jacobian of an elliptic curve E over the complex numbers.

This latter arises when one reduces zero-degree divisors of the curve modulo the equivalence relation introduced by Rosenlicht in 1954: two zero-degree divisors are equivalent, if and only if their difference is a principal divisor div(f), with the property that f(M) = f(N) for two fixed suitable points M and N on the elliptic curve.  Such a generalized Jacobian turns out to be the extension of the multiplicative group of the ground field by the classic Jacobian of the curve, more or less the same way C turns out to be the extension of the ring of integers R by the finite Jacobian of C modulo k.

Mikel Garciarena Perez

The lower central series of certain Grigorchuk-Gupta-Sidki groups

Groups of automorphisms of regular rooted trees are a rich source of examples with interesting properties in group theory, and they have been used to solve very important problems.  The first Grigorchuk group, defined by Grigorchuk in 1980, is one of the first instances of an infinite finitely generated periodic group, thus producing a negative solution to the General Burnside Problem.  It is also the first example of a group with intermediate growth, hence solving the Milnor Problem.  Many other groups of automorphisms of rooted trees have since been defined and studied.  Important examples are the Gupta-Sidki p-groups, for p an odd prime, and the second Grigorchuk group.  These are again finitely generated infinite periodic groups and they belong to the large family of the so-called Grigorchuk-Gupta-Sidki groups (GGS-groups, for short).

Some research has been done regarding the lower central series of groups of automorphisms of regular rooted trees and more specifically of some particular GGS-groups, but the knowledge is scarce.  The aim of this talk is to present some of the techniques and tools that we have developed to understand the lower central series of some of the GGS-groups that act on the p-adic tree, and how do we use them to completely determine the lower central series of some GGS-groups such as the p-Fabrykowski-Gupta groups.

This is a joint work with Gustavo Fernandez-Alcober and Marialaura Noce.

Ioana-Claudia Lazar

Minimal displacement set for weakly systolic complexes

We show that the minimal displacement set in a weakly systolic simplicial complex embeds isometrically into the complex and that it is systolic.  As corollaries, we prove that any isometry of a weakly systolic complex either fixes the barycentre of some simplex (elliptic case) or it stabilizes a thick geodesic (hyperbolic case).  We investigate the structure of the minimal displacement set in an 8-located simplicial complex with the SD'-property.  We show that such set embeds isometrically into the complex.  A simplicial complex is 8-located if it is flag and every full homotopically trivial loop of length at most 8, is contained in a 1-ball.

Francesco Milizia

Davis' manifolds with positive simplicial volume

The simplicial volume is a homotopy invariant of manifolds; this talk is about the simplicial volume of a specific class of manifolds: those obtained from Davis’ reflection group trick. When attempting to understand which ones have positive simplicial volume, we are led to problems about triangulations of spheres and simplicial maps between them. A connection with the theory of graph minors will also be presented.

Piotr Mizerka

Induction of spectral gaps for the cohomological Laplacians of SL(n,ℤ) and SAut(F_n) 

Bader and Nowak introduced cohomological Laplacians whose properties are linked to vanishing of group cohomologies.  In particular, for a group G, the degree one Laplacian possessing a positive spectral gap is equivalent to G satisfying Kazhdan's property (T).  The latter statement is a consequence of the works of Bader, Nowak and Sauer.  On the other hand, due to Ozawa, it had been known before that property (T) is equivalent to the existence of a positive spectral gap related to the degree zero Laplacian.  Kaluba, Kielak and Nowak applied the characterization of Ozawa to show property (T) for the automorphism groups of free groups.  They developed an induction technique for that purpose: from the existence of a spectral gap for the automorphism group of smaller degree, they were able to deduce the spectral gap for higher degree automorphism groups.

In this talk, we describe a method of adapting the induction technique of Kaluba, Kielak and Nowak to the degree one Laplacian.  As an application of our method, we provide an alternative proof of property (T) for the group of n-by-n matrices with determinant one for n ≥ 3.

Andoni Zozaya

On the generation of classical simple Lie algebras

The problem of generating simple algebraic structures has been extensively studied.  For instance, finite simple groups are 2-generated, and, furthermore, they are 3/2-generated, meaning that for every non-trivial element x in G there exists another element y in G such that x and y together generate G.

In this talk, we will discuss these generation properties for the Lie algebras of traceless matrices over fields of positive characteristic.  We will show how to obtain many pairs of generators for these Lie algebras, and we will provide examples of finite dimensional simple Lie algebras that are not generated by two elements, with these peculiarities arising in characteristics 2 and 3.  Joint work with Cantor and Jezernik.