You may also be interested in the Bielefeld-Münster Seminar on Groups, Geometry and Topology

Or the Colloquium by Alex Lubotzky (Hebrew University) on 18/06/2015 Or the Colloquium by Jon McCammond (UCSB) on 2/07/2015

Abstract: Associated to a twin building there is an opposition complex. I will talk about research in progress concerning their connectivity properties and possibly sketch how to use it to deduce isomorphisms in group homology.

Abstract: In this talk, I will discuss a procedure using the action on homology of covers for obtaining infinitely many "virtual" arithmetic quotients of mapping class groups of closed surfaces, (surjective maps up to finite index). Specifically, for any irreducible rational representation of a finite group of rank less than g, we produce a corresponding virtual arithmetic quotient of the genus g mapping class group. Particular choices of irreducible representations of finite groups yield arithmetic quotients of type Sp(2m), SO(2m, 2m), and SU(m, m) for arbitrarily large m in every genus. Joint with F. Grunewald, M. Larsen, and A. Lubotzky.

Abstract: Let X=SxExB be the metric product of a symmetric space S of non-compact type, a Euclidean space E and a product of Euclidean buildings B. Let L be a discrete group acting isometrically and cocompactly on X. I will discuss a family of quasi-isometry invariants for such L, namely the k-dimensional Dehn functions which measure the difficulty to fill k-spheres by (k+1)-balls (for k=1,2,…,dim X-1). Basic examples of groups L as above are uniform S-arithmetic subgroups of reductive groups defined over global fields. I will also discuss a mostly conjectural picture for non-uniform S-arithmetic groups.

Abstract: Let V be a finite faithful irreducible G-module for some finite group G. A base B is a subset of V with the property that the joint stabilizer in G of all vectors in B is the identity. The minimal size of a base is denoted by b(G). There are many results on b(G) and such theorems, in case G is solvable, have applications to Gluck's conjecture. This is the following. Let H be a finite solvable group and let F(H) be its Fitting subgroup (the largest nilpotent normal subgroup of H). Then |H:F(H)| is at most the square of the largest complex irreducible character degree of H. The first part of my talk (minimal base sizes) will be on a recent work of mine with Zoltan Halasi. The second part (on Gluck's conjecture) is joint work with James P. Cossey, Zoltan Halasi and Hung Ngoc Nguyen.

Abstract: For a locally compact group G, the Howe-Moore property is the vanishing at infinity of all matrix coefficients of the unitary representations of G that are without non-zero G-invariant vectors. Apart from connected, non-compact simple real Lie groups, isotropic simple algebraic groups over non Archimedean local fields and closed, topologically simple subgroups of Aut(T) that act 2-transitively on the boundary of T, where T is a bi-regular tree of valence ≥ 3, there are no other known examples. If we only restrict to the case of a d-regular tree T

ABSTRACT:

Let G be a group, let H be a subgroup of G and let V be an irreducible KG-module over a field K. We say that (G,H,V) is an irreducible triple if the restriction of V to H is irreducible.

Irreducible triples arise naturally in the investigation of maximal subgroups of simple groups (both finite and algebraic). In this context, their study can be traced back to work of Dynkin in the 1950s on the subgroup structure of the classical algebraic groups over the complex numbers. Through more recent work of Seitz, Testerman and others, Dynkin's analysis has been extended to all simple algebraic groups over any algebraically closed field. Indeed, the problem of determining the irreducible triples (G,H,V) for simple algebraic groups has essentially been reduced to the case where G is a classical group and H is disconnected.

In this talk I will report on recent work towards a complete classification of the irreducible triples (G,H,V) when G is a classical algebraic group and H is disconnected and infinite. I will use some examples to illustrate the main ideas, and I will briefly discuss some related problems.

This is joint work with Soumaia Ghandour, Claude Marion and Donna Testerman.

Abstract: Rank and Deficiency gradients quantify the asymptotics of finite approximations of a group. These group invariants have surprising connections with many different areas of mathematics: 3-manifolds, L

Abstract: Let & Gamma; be a finite simple graph with vertex set S. The associated right-angled Coxeter group W

Abstract: In order to understand the geometry of a manifold, it is useful to analyse its finite sheeted covers. The field of homology growth is concerned with the behaviour of homological invariants, such as Betti numbers, in families of finite sheeted covers. We give an introduction to homology growth from a group theoretic point of view; we focus on the behaviour of Betti numbers in towers of finite index subgroups. We indicate relations to L

Abstract: For a path algebra of type A

Abstract: A quasi-geodesic g is called Morse if any (L,C)-quasi-geodesic connecting any two points on g stays C=C(L,C) close to g. In many cases in groups, these Morse quasi-geodesics are periodic, that is, there exists an element x in the group such that <x> is a Morse quasi-geodesic. It is well-known that for example (relatively) hyperbolic groups or the mapping class group have periodic Morse quasi-geodesics. I will present a construction of such quasi-geodesics in a torsion group, hence providing first examples of groups that have Morse quasi-geodesics but no Morse elements. This is work in progress with Romain Tessera.

Abstract: In 1997 Ivanov proved that the automorphism group of the complex of curves associated to a surface S is isomorphic to the extended mapping class group Mod(S), for most surfaces. His work generated a flurry of activity, with similar results obtained by several different authors for various other simplicial complexes associated to surfaces. Ivanov then posed a "metaconjecture" stating that every "sufficiently rich" complex associated to a surface S has Mod(S) as its group of automorphisms. In this talk we will discuss a resolution of Ivanov's metaconjecture for a wide class of complexes. This is joint work with Dan Margalit.

Abstract: The motivation behind this work is the investigation on the unit group of an order O in a rational group ring ℚG of a finite group G. In particular we are interested in getting a presentation in terms of generators and relations of the unit group of ℤG. By the Wedderburn-Artin Theorem, the study of U(ℤG) may be reduced, up to commensurability, to the study of SL

Abstract:

I will discuss an ongoing project with H. Van Maldeghem concerning (exceptional) algebraic groups and their associated geometries, one of our motives being to obtain a geometric construction of the 248-dimensional E8-module.

The main goal is to give a uniform axiomatic description of the embeddings in projective space of the varieties corresponding with the geometries of exceptional Lie type over arbitrary fields. This comprises a purely geometric characterization of F4, E6, E7 and E8.

Abstract: Dual braid monoids associated to finite Coxeter groups are Garside monoids, providing a replacement of the positive braid monoid with a new set of generators corresponding to the whole set of reflections of the Coxeter group. As Garside monoids, they possess a finite set of so-called simple elements, in canonical bijection with noncrossing partitions. They imbed into their group of fractions, which is isomorphic to the Artin-Tits group, and this isomorphism is poorly understood (the proof is case-by-case). Using the noncrossing models, we will show positivity properties of the linear expansions of the simple elements in the canonical basis of the Hecke algebra in types A and B (joint with François Digne); this require to view the simple elements in the Artin-Tits group, hence a better understanding of the isomorphism above, and also relies and positivity theorems showed using categorification.

Given a finitely generated profinite group G, consider the series P

Abstract: A discrete group of circle homeomorphisms is a Fuchsian group if and only if it is a convergence group (this is due to Tukia, Casson-Jungreis, Gabai, ...). We show that the convergence property can also be characterized in terms of invariant laminations on the circle, so this gives a new characterization of Fuchsian groups. We also discuss what can be said about fibered hyperbolic 3-manifold groups. The main motivation of the work is Thurston's universal circle theory.

Abstract:

Consider SL(n,R), the symmetric space X

Generalizing reduction theory to arithmetic groups, Godement found an adelic formulation treating all places simultaneously. Let K be a global number field, G be a semisimple group (think of SL(n,-)), and let A be the ring of adeles of K. Then G(K) is discrete in G(A), and Godement finds a fundamental domain (coarsly) for the action of G(K) on G(A). Later, Behr and Harder transferred this to the case when K is not a global number field, but a function field.

In 2012 Bux-Kohl-Witzel gave a geometric reformulation of Behr-Harder for the S-arithmetic case. They introduced a so called "reduction datum". Now the natural question arises, whether this geometric reformulation can be transferred back to the case of a number field: Yes it can be transferred back! In this talk I explain the geometry and state the basic steps of the proof.

Abstract: I will describe a construction of 'quasi-arithmetic' discrete groups acting with finite co-volume on hyperbolic space: these lattices are non-arithmetic but in some sense close to being arithmetic. I will also explain how these groups are distinguished from the well-known non-arithmetic lattices of M. Gromov and I. Piatetski-Shapiro (1986), which are not quasi-arithmetic.

Abstract: here

Abstract: I will prove two conjectures, by Klopsch-Voll and Stasinski-Voll, about the signed distribution of a new statistic, odd length, defined on Weyl groups of type A and B. I will also define a natural analogue of this statistic for Coxeter groups of type D and prove some results about its signed generating function. Finally I will give other algebraic and geometric interpretations of the odd length. This is joint work with Francesco Brenti.

Abstract: Expander graphs have played, in the last few decades, an important role in computer science, and in the last decade, also in pure mathematics. In recent years a theory of "high-dimensional expanders" is starting to emerge - i.e., simplical complexes which generalize various properties of expander graphs. This has some geometric motivations (led by Gromov) and combinatorial ones (started by Linial and Meshulam). The talk will survey the various directions of research and their applications, as well as potential applications in math and CS. Some of these lead to questions about buildings and representation theory of p-adic groups. (We will survey the work of a number of people. The works of the speaker in this direction in recent years are with Tali Kaufman, David Kazhdan and Roy Meshulam).

Abstract: TBA

Abstract: We use techniques from geometric group theory, from the geometry of Coxeter complexes and from representation theory to calculate dimensions and prove nonemptiness of affine Deligne--Lusztig varieties X_x(b) in the case that b is a pure translation. Here G is a reductive group over a field F and W the affine Weyl group of G. The associated varieties X_x(b) are indexed by elements b in G and x in W. Our approach is constructive and type-free, sheds new light on the reasons for existing results and reveals new patterns. One of the main tools used are the root operators on folded galleries which were introduced by Peter Littelmann and Stephane Gaussent. Moreover, as an application, we are able to obtain the first exact calculation of reflection length in affine Weyl groups for elements other than pure translations.

Abstract: Whitehead groups, and more generally higher algebraic K-theory groups of group rings, play an important role in the study of high-dimensional manifolds. I will present joint work with Wolfgang Lück, Holger Reich, and John Rognes [arXiv:1504.03674], in which we proved that, after rationalization, the Whitehead group of any group G satisfying mild homological finiteness assumptions contains the colimit of the Whitehead groups of all finite subgroups of G. I will also show how together with Ross Geoghegan [arXiv:1401.0357] we used this to prove that the Whitehead group of Thompson's group T is infinitely generated. I will focus on the finiteness assumptions on the groups that our methods require, and explain how these results are consequences of more general theorems that we proved about the Farrell-Jones Conjecture in algebraic K-theory, generalizing a seminal theorem of Bökstedt-Hsiang-Madsen, and how they relate to the Leopoldt-Schneider Conjecture in algebraic number theory.

Abstract: Braiding strings and looking at reflections in mirrors are common activities that lead to complicated mathematical structures. The symmetry groups of the Platonic solids (and all higher dimensional regular polytopes) are finite groups generated by reflections. Discrete groups generated by reflections also are at the heart of the theory of Lie groups and Lie algebras. In the early 1960s Jacques Tits introduced the notion of a general Coxeter group that encompasses both of these contexts using elegant group presentations, a class that also include reflection groups acting on hyperbolic spaces and other exotic contexts. There is second class of groups called Artin groups that are related to Coxeter groups in much the same way that Artin's braid groups are related to the symmetric groups. In this talk I will survey a variety of results about such groups with a particular emphasis on the reflection groups (Coxeter groups) and generalized braid groups (Artin groups) that act on and are associated with spheres and euclidean spaces.

Abstract: This is a joint work with Gili Golan. We answer several questions by Vaughan Jones about a subgroup of F which encodes all oriented links.

Abstract: Es soll gezeigt werden, dass sich aus den Kohomologiegruppen von GL

Abstract: Thurston introduced train tracks and geodesic laminations as tools to study surface diffeomorphisms and Kleinian groups. We'll start the talk with a relaxed introduction to these. Then, in analogy with the end invariants of Kleinian groups and Teichmüller geodesics, we will define the end invariants of an infinite splitting sequence of train tracks. These end invariants determine the set of laminations that are carried by all tracks in the infinite splitting sequence. If there is time, we'll use these ideas to sketch a new proof of Klarreich's theorem, determining the boundary of the curve complex.

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