with Applications Discrete Groups and Geometric Structures · rijk workshop on Discrete Groups and...

Conference Booklet of the 5th Kortrijk Workshop

Discrete Groups and Geometric Structureswith Applications

K.U.Leuven Campus Kortrijk,May 26 - May 30, 2008

Frank De ClerckYves FelixWilliam GoldmanFritz GrunewaldPaul IgodtAlain ValetteScientific Committee

Karel DekimpeYves FelixPaul IgodtNansen PetrosyanStefaan VaesOrganizing Committee

Dear participant,

We wish you a warm welcome at the fifth edition of the Kort-rijk workshop on Discrete Groups and Geometric Structures,with Applications.

The conference is taking place exactly at the spot where itstarted in its first edition in 1996. We are grateful to you allfor being here, and in particular to the invited speakers whoaccepted to outline this conference with their inspiring talksand research.

2008 will be a year to remember for mathematicians activein group theory and geometric structures, as this year’sAbel prize has been awarded to 2 eminent colleagues whoperformed their research mainly in an area which is hard todescribe shorter than in the 5 words: “Discrete Groups andGeometric Structures”.

We invite you all to join the conference in a pleasant atmo-sphere of friendship and collaboration.

Karel Dekimpe and Paul Igodt

Abel Prize 2008

Awarded to John Thompson and Jacques Tits

May 20, 2008

John Griggs Thompson Jacques TitsGraduate Research Professor, Professor Emeritus,

University of Florida College de France

for their profound achievements in algebra andin particular for shaping modern group theory

Theorem (Feit – Thompson, 1962). Every finite group of odd order is soluble.

THE ABEL PRIZE COMMITTEE – 2008. Thompson revolutionised the theoryof finite groups by proving extraordinarily deep theorems that laid the founda-tion for the complete classification of finite simple groups, one of the greatestachievements of twentieth century mathematics.

Theorem (Tits, 1972). Every finitely generated linear group is either virtuallysolvable or contains a copy of the free group on two generators.

THE ABEL PRIZE COMMITTEE – 2008. Tits created a new and highly influen-tial vision of groups as geometric objects. He introduced what is now knownas a Tits building, which encodes in geometric terms the algebraic structureof linear groups.

Contents

Welcome 2

Contents 4

1 Schedule 5

2 Invited talks 11

3 Plenary talks 15

4 Short Communications 19

List of Participants 33

Thanks 40

4

Schedule 1The conference is scheduled with 8 invited one-hour talks, presented by themain invited speakers, i.e.

• Jørgen Andersen (University of Aarhus),

• Martin Bridson (Oxford University),

• Benson Farb (University of Chicago),

• Fritz Grunewald (Heinrich-Heine Universitat Dusseldorf),

• Ursula Hamenstadt (Rheinische Friedrich-Wilhelms-Universitat Bonn),

• Alex Lubotzky (Hebrew University of Jerusalem),

• Nicolas Monod (Ecole Polytechnique Federale Lausanne) and

• John Parker (University of Durham).

The conference schedule also contains 6 plenary 45-minutes talks, and 3125-minutes talks in parallel sessions.

5

6 Conference booklet

Monday, May 26

Morning session

08.30-09.00 Hall B Registration.09.00-09.15 B422 Welcome and Opening of the Workshop by P. Igodt.09.15-10.15 B422 Nicolas Monod Abstract page 13

On the structure of isometry groups in non-positive cur-vature.

10.15-10.45 Hall B Coffee break10.45-11.45 B422 Ursula Hamenstadt Abstract page 12

Subgroups of Out(Fn).

Afternoon session

14.00-14.25 B422 Thierry Barbot Abstract page 20Quasi-Fuchsian representations in SO(2,n).

A305 Rafal Lutowski Abstract page 27On symmetry of flat manifolds.

14.35-15.00 B422 Dietrich Burde Abstract page 21Degenerations of (pre)-Lie algebras.

A305 Raquel Agueda Mate Abstract page 19Homotopy classes of curve systems on the twice puncturedtorus.

15.10-15.35 B422 Sandra Deschamps Abstract page 23Simply transitive NIL-affine actions and LR-structures.

A305 Habib Marzougui Abstract page 28Classification of Exotic circles of Pr(S1).

15.40-16.10 Hall B Coffee break16.10-16.35 B422 Kim Vercammen Abstract page 31

LR-structures on nilpotent Lie algebras.A305 Mathieu Carette Abstract page 22

The automorphism group of accessible groups and of hy-perbolic groups.

16.45-17.30 B422 Bernhard Muhlherr Abstract page 16The geometries and lattices associated with RGD-flips.

17.45h Shuttle

Schedule 7

Tuesday, May 27Morning session

9.00h – 10.00h B422 Jørgen Andersen Abstract page 11Mapping Class Groups do not have Kazhdan’s Prop-erty (T).

10.00h – 10.30h Hall B Coffee Break10.30h – 11.30h B422 Benson Farb Abstract page 12

Isometries, rigidity, and universal coverings.11.40h – 12.05h B422 Eric Freden Abstract page 24

Growth in Baumslag-Solitar groups: subgroups andrationality.

A305 Leandro Cagliero Abstract page 22Faithful representations of minimal dimension of cur-rent Heisenberg Lie algebras.

Afternoon session

14.00h – 14.25h B422 Udo Baumgartner Abstract page 20A Bieberbach Theorem for Automorphism Groups ofTotally Disconnected, Locally Compact Groups.

A305 Michelle Bucher-Karlsson Abstract page 21Milnor-Wood inequalities for manifolds locally isomet-ric to products of hyperbolic planes.

14.35h – 15.00h B422 Virginie Charette Abstract page 23Groups of affine transformations generated by inver-sions.

A305 Cristina Hretcanu Abstract page 25On some properties of submanifolds in golden Rieman-nian manifolds .

15.10h – 15.35h B422 Mohammed Guediri Abstract page 24Global hyperbolicity of left-invariant Lorentz metricson Lie groups.

A305 Leonid Potyagailo Abstract page 30Floyd Map and Convergence Groups.

15.40h – 16.10h Coffee Break16.10h – 16.35h B422 Marek Halenda Abstract page 25

Properties of Kahler flat manifolds.A305 Elena Klimenko Abstract page 26

The geometry of Kleinian groups with real parameters.16.45h – 17.30h B422 Stefaan Vaes Abstract page 17

Actions of the free group on infinitely many genera-tors providing factors and equivalence relations withprescribed fundamental group.

17.45h Shuttle


Wednesday, May 28

Morning session

9.00h – 10.00h B422 Alex Lubotzky Abstract page 13Linear representations of the automorphism groupsof free groups.

10.00h – 10.30h Hall B Coffee Break10.30h – 11.15h B422 Lizhen Ji Abstract page 15

Rigidity and duality properties of arithmetic groups,mapping class groups and outer automorphismgroups of free groups.

11.25h – 12.10h B422 Pierre-Emmanuel Caprace Abstract page 15Quasi-characters of groups acting on buildings.

Afternoon session

13.30h Conference Picture at the Campus Restaurant

14.30h Departure of excursion

15.00h Guided visit “Le Vieux Lille”

18.30h Short sight seeing stop at Ypres

19.30h Conference Dinner at Mount Kemmel

23.00h Return to Kortrijk

Schedule 9

Thursday, May 29

Morning session

9.00h – 10.00h B422 John Parker Abstract page 13Complex Hyperbolic Lattices.

10.00h – 10.30h Hall B Coffee Break10.30h – 11.30h B422 Fritz Grunewald Abstract page 12

Cohomology of Bianchi groups.11.40h – 12.05h B422 Gilbert Levitt Abstract page 26

Counting growth types of automorphisms of freegroups.

A305 Karin Melnick Abstract page 28Conformal flows on compact pseudo-Riemannianmanifolds.

Afternoon session

14.00h – 14.25h B422 Daniel Appel Abstract page 19On the index of congruence subgroups of Aut (Fn).

A305 Bartosz Putrycz Abstract page 31Spin structures on flat 4-manifolds.

14.35h – 15.00h B422 Martin Lustig Abstract page 27Trees versus currents, for automorphisms of freegroups.

A305 Seonhee Lim Abstract page 27Volume entropy for hyperbolic buildings.

15.10h – 15u.35h B422 Pieter Penninckx Abstract page 29The Anosov theorem for infra-nilmanifolds with a2-perfect holonomy group.

A305 Yoshinobu Kamishima Abstract page 25Seifert fibered structure and rigidity on real Botttowers.

15.35h – 15.55h Hall B Coffee Break15.55h – 16.20h B422 Koen Struyve Abstract page 31

Twinnings of R-buildings.A305 Wolfgang Moens Abstract page 29

Faithful representations of minimal degree for re-ductive Lie algebras.

16.30h – 16.55h B422 Nansen Petrosyan Abstract page 30Homology of Hantzsche-Wendt Groups.

A305 Adara-Monica Blaga Abstract page 21Canonical connections on k-symplectic manifoldsunder reduction.

17.00h Shuttle


Friday, May 30

Morning session

9.00h – 10.00h B422 Martin Bridson Abstract page 11Actions of automorphism groups of free groups.

10.00h– 10.25h Hall B Coffee Break10u25h– 10.50h B422 Ilya Kapovich Abstract page 26

Analogues of curve complex for free groups.11.00h – 11.45h B422 Pierre de la Harpe Abstract page 15

Spaces of closed subgroups of locally compact groups.11.55h – 12.40h B422 Karen Vogtmann Abstract page 17

Automorphisms of right-angled Artin groups.12.40h B422 Closure of the conference

13.00h Shuttle

Invited talks 2Mapping Class Groups do not have Kazhdan’s Property (T)

Jørgen Andersen – Aarhus Universitet

We prove that the mapping class group of a closed oriented surface of genusat least two does not have Kazhdan’s property (T). The Hilbert space repre-sentation needed is the completion of a direct sum of a certain subset of theReshetikhin-Turaev TQFT representations of the mapping class groups. Theserepresentations are known by the work of Justin Roberts to not have any fixedvectors. We construct an almost fixed vector by a coherent state constructionin the geometric approach to these representations. In particular we find thatthe coherent states are asymptotic flat with respect to the Hitchin connection,and we use this to prove that they are almost fixed.

Scheduled on Tuesday morning.

Actions of automorphism groups of free groups

Martin Bridson – University of Oxford

Joint work with Karen Vogtmann

Let F be a finitely generated free group of rank at least 3 and let SAut(F ) de-note its unique subgroup of index two. The standard linear action of SL(n,Z)on Euclidean n-space Rn induces non-trivial actions of SAut(F ) on Rn and onthe (n−1)-sphere. In this talk I shall explain joint work with Karen Vogtmannin which we prove that SAut(F ) cannot act non-trivially by homeomorphismson acyclic (homology) manifolds or (generalized) spheres of smaller dimen-sion, and hence neither can SL(n,Z).

The proof involves Smith theory (which I shall explain) and an under-standing of the torsion in SAut(F ).

If time allows, I’ll switch attention to non-positively curved spaces andexplain what is known about the question: what is the least dimension inwhich these (and related) groups can act on a complete CAT(0) space without

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a global fixed point? I shall present a partial answer based on a generalisationof Helly’s theorem and on an understanding of the lattice of finite subgroupsin Aut(F ). Scheduled on Friday morning.

Isometries, rigidity, and universal coverings

Benson Farb – University of Chicago

Joint work with Shmuel Weinberger

Which contractible Riemannian manifolds (e.g. Rn with any metric) coverboth compact and noncompact, complete finite volume manifolds? Which Rie-mannian products X×Y cover compact nonproducts? Do complex manifoldsM with c1(M) < 0 holomorphically split as a product of a locally symmet-ric manifold and a “rigid” manifold? What are the isometries of Teichmullerspace? These seemingly unrelated questions turn out to be instances of a sin-gle underlying phenomenon, allowing for complete answers to each question.The goal of this talk will be to explain some of the ideas behind this, which isjoint work with Shmuel Weinberger.


Cohomology of Bianchi groups

Fritz Grunewald – Heinrich-Heine-Universitat Dusseldorf

I will report on new results and conjectures about the cohomology of discretesubgroups of PSL(2,C).

Scheduled on Thursday morning.

Subgroups of Out(Fn)

Ursula Hamenstadt – Rheinische Friedrich-Wilhelms-Universitat Bonn

Let Fn be the free group with n ≥ 2 generators, and let Out(Fn) be the outerautomorphism group of Fn. Using a recent result of Kapovich and Lustig, weshow that every homomorphism of an irreducible lattice in a semi-simple Liegroup of non-compact type into Out(Fn) has finite image. This extends anearlier result of Bridson and Farb for non-uniform lattices.

Scheduled on Monday morning.

Invited talks 13

Linear representations of the automorphism groups of free groups

Alex Lubotzky – Hebrew University of Jerusalem

Joint work with Fritz Grunewald

Let F = Fn be the free group on n generators and A =Aut(Fn). The action ofA on the abelianization of F gives a representationR0 ofA onto the arithmeticgroup GL(n,Z). We study the action of A (or finite index subgroups of it) onthe abelianizations of finite index normal subgroups L of F , with F/L = G,i.e., on the relation module of G. Using Gashutz theory of relation modules,we produce many new representations of A onto a rich class of arithmeticgroups (where R0 corresponds to the case G being the trivial group).

Scheduled on Wednesday morning.

On the structure of isometry groups in non-positive curvature

Nicolas Monod – Ecole Polytechnique Federale de Lausanne

Joint work with Pierre-Emmanuel Caprace

We investigate the general properties of isometry groups of CAT(0) spacesand of their discrete subgroups. Particular emphasis is on the analogies orcontrasts with the theory of semi-simple groups over local fields and arith-metic groups.

Scheduled on Monday morning.

Complex Hyperbolic Lattices

John Parker – University of Durham

There are relatively few examples of complex hyperbolic lattices known. Thetwo major classes of examples both have their origin in work of Picard in the1880’s. First, there are lattices coming from the ring of integers in an imag-inary quadratic number field and, more generally, lattices which arise fromarithmetic constructions. Secondly, there are lattices that are monodromygroups of certain hypergeometric functions. These include the examples ofnon-arithmetic lattices constructed by Mostow. Other constructions yield lat-tices commensurable with one of those described above. In this talk I willdiscuss the geometry of these lattices and their action on complex hyperbolicspace via examples.


Plenary talks 3Quasi-characters of groups acting on buildings

Pierre-Emmanuel Caprace – Institut des Hautes Etudes Scientifiques

Joint work with Koji Fujiwara

We describe a construction of quasi-characters for groups admitting stronglytransitive actions on irreducible non-Euclidean (possibly non locally compact)buildings. This yields examples of finitely presented simple groups of infinitecommutator width.


Spaces of closed subgroups of locally compact groups

Pierre de la Harpe – Universite de Geneve

The set C(G) of closed subgroups of a locally compact group G has a naturaltopology which makes it a compact space. This topology has been defined invarious contexts by Vietoris, Chabauty, Grigorchuk, and many others.

The talk will describe the space C(G) in a few elementary cases, for Gthe complex plane, in which case C(G) is a 4-sphere (a result of Hubbardand Pourezza), and for the 3-dimensional Heisenberg group H, in which caseC(H) is a 6-dimensional singular space recently investigated by M. Bridson,V. Kleptsyn and P.H.

Scheduled on Friday morning.

Rigidity and duality properties of arithmetic groups, mappingclass groups and outer automorphism groups of free groups

Lizhen Ji – University of Michigan

It is known that the three important classes of groups in the title share manycommon properties. In this talk, I will explain some more results which willillustrate further their unity. For example, a generalization of the Mostow

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strong rigidity for the latter two classes of groups, Poincare duality vs. dualityproperties for them, and other related quasi-isometry rigidity results etc.

If time permits, I will also explain a common extension property for holo-morphic maps on the punctured disc to the Deligne-Mumford compactificationof the moduli space of Riemann surface (i.e., the quotient of the Teichmullerspace by the mapping class group), and the Baily-Borel compactificaiton ofarithmetic Hermitian locally symmetric spaces (the latter is the known Boreextension theorem, and important for the theory of variation of Hodge struc-tures).


The geometries and lattices associated with RGD-flips

Bernhard Muhlherr – Universite Libre de Bruxelles

Joint work with A. Devillers, R. Gramlich, M. Horn and P.-E. Caprace

RGD-systems have been introduced by Tits in 1987 in order to investigate Kac-Moody groups over fields. The latter are infinite-dimensional generalisationsof Chevalley-groups. To each RGD-system there is naturally associated a twinbuilding which turns out to be very useful for the understanding of thesegroups. In fact, the notions of an RGD-system and a Moufang twin buildingare equivalent.

A flip of an RGD-system is an automorphism switching the two halves ofits associated twin building. It turns out that Borel groups and centralizersof flips behave similarly. There is a natural geometry associated with both.In the case of Borel groups this geometry has been studied intensively andit has been used by Abels and Abramenko to study finiteness properties ofcertain arithmetic groups. It was observed independently by B. Remy andCabone/Garland that Borel groups of locally finite RGD-systems are latticesin the full automorphism group of the building associated with it. It turns outthat the same is true for centralizers of flips. In this way we get a new class ofhigher rank lattices of ‘algebraic origin’.

In my talk I intend to explain the analogy between Borel groups and cen-tralizers of flips, to present some results about the geometry of flips with ap-plications and to explain why centralizers of flips yield lattices. The resultsabout the geometries have been obtained in joint work with A. Devillers, R.Gramlich and M. Horn, whereas the ‘lattice-part’ of my talk is about researchin progress with P.-E. Caprace and R. Gramlich.

Scheduled on Monday afternoon.

Plenary talks 17

Actions of the free group on infinitely many generators providingfactors and equivalence relations with prescribed fundamental

group

Stefaan Vaes – Katholieke Universiteit Leuven

Joint work with Sorin Popa

Murray and von Neumann introduced in 1943 an invariant for factors and forcountable equivalence relations, called fundamental group. It is a subgroupof the positive real numbers R+. We exhibit a family S of subgroups of R, con-taining R itself, all its countable subgroups, as well as uncountable subgroupsof any Hausdorff dimension. For any group H in S, we construct free ac-tions of the free group on infinitely many generators yielding factors and orbitequivalence relations whose fundamental group is exp(H). This provides inparticular the first separable factors and equivalence relations with uncount-able fundamental group different from R+. Part of my talk will be devoted toa general audience introduction to II1 factors and orbit equivalence relations.

Scheduled on Tuesday afternoon.

Automorphisms of right-angled Artin groups

Karen Vogtmann – Cornell University, Ithaca

Right-angled Artin groups (RAAG’s) interpolate between free groups and freeabelian groups: they are described by giving a set of generators, some of whichcommute. A RAAG is conveniently described by drawing a graph whose ver-tices are the generators, with an edge between two generators if and only ifthey commute. Our focus is on understanding to what extent automorphismgroups of RAAG’s behave like automorphism groups of free groups and offree abelian groups. I will describe new results on finiteness properties of thegroup of outer automorphisms of a RAAG, including the case when the asso-ciated graph is a tree. This case is of special interest because the associatedRAAG is a 3-manifold group.


Short Communications 4Homotopy classes of curve systems on the twice punctured torus

Raquel Agueda Mate – Universidad de Castilla - La Mancha

Let M be a hyperbolic 3-manifold obtained from a genus-2 handlebody bypinching a non-dividing curve. We parametrize the space of ∂M -homotopyclasses of curve systems, free homotopy classes of families of disjoint essentialsimple loops on the twice punctured torus ∂M . We use the p-coordinates,p(γ)

= (q1, q2, p1, p2) ∈ (Z+ ∪ {0})2 × Z2, a variant on Dehn-Thurston co-ordinates developed by L. Keen, J. Parker and C. Series in [1] (see [2] or [3]for a different approach).

As M has compressible boundary ∂M , there are curve systems in M whichare not homotopic in ∂M while they are homotopic in M : the curve systemsbelong to a different ∂M -homotopy class, but to the same M -homotopy class.We find a representative curve system γ0 in each M -homotopy class, uniqueup to ∂M -homotopy, so that its p-coordinates fulfil the following conditions:

q1 ≤ q2 and 0 ≤ p2 < q2.

[1] L. Keen, J. Parker, C. Series: Combinatorics of simple closed curves on the twicepunctured torus. Israel J. Math. 112 (1999), 29-60.

[2] F. Luo, R. Stong: Dehn-Thurston coordinates for curves on surfaces. Commun.Anal. Geom. 12 (2004), 1-41.

[3] R. C. Penner, J. L. Harer: Combinatorics of Train Tracks. Ann. of Math. Studies125, Princeton University Press, 1992.


On the index of congruence subgroups of Aut (Fn)

Daniel Appel – Heinrich-Heine-Universitat Dusseldorf

For an epimorphism π : Fn → G of the free group Fn onto a finite group G wecall Γ(G, π) = {ϕ ∈ Aut(Fn) | πϕ = π} the standard congruence subgroup

19


of Aut(Fn) associated to G and π. In the case n = 2 we present formulasfor the index of Γ(G, π) where G is abelian or dihedral. Moreover, we showthat congruence subgroups associated to dihedral groups provide a family ofsubgroups of arbitrary large index in Aut(F2) generated by a fixed numberof elements. This implies that finite index subgroups of Aut(F2) cannot bewritten as free products.

Scheduled on Thursday afternoon.

Quasi-Fuchsian representations in SO(2,n)

Thierry Barbot – Ecole Normale Superieure de Lyon

Let Γ be a (torsion-free) uniform lattice in SO(1, n) (n ≥ 2). The natural in-clusion SO(1, n) ⊂ SO(2, n) induces representations ρ0 : Γ → SO(2, n) calledFuchsian (according to Mostow rigidity, such a represention is unique up toconjugacy if n ≥ 3). In this talk, I will present the proof of the followingresult:

Theorem. Every representation ρ : Γ → SO(2, n) which can be continuouslydeformed to a Fuchsian representation is faithfull, discrete and purely loxodromic(i.e. for every γ in Γ, ρ(γ) admits one real eigenvalue λ such that λ > 1).

The proof relies on a deep analysis of the geometry of anti-de Sitter spaceAdSn+1, which is the Lorentzian analog of hyperbolic space Hn, and in partic-ular to its causal properties. It admits in particular a natural conformal bound-ary, called Einstein universe Einn. The theorem above is a corollary of the mostprecise result: every deformation of a Fuchsian representation into SO(2, n) isquasi-Fuchsian, i.e. preserves a closed acausal quasi-sphere in Einn.


A Bieberbach Theorem for Automorphism Groups of TotallyDisconnected, Locally Compact Groups

Udo Baumgartner – University of Newcastle

Joint work with Gunter Schlichting and George Willis

Automorphisms of a totally disconnected, locally compact group G may beanalyzed by means of their action on the set B(G) of compact, open subgroupsof G. The function d(V,W ) := log

(|V : V ∩W | · |W : W ∩ V |

)defines a metric

on B(G) and important invariants of groups of automorphisms of G can bedescribed in terms of the action of the induced group of isometries of themetric space B(G).

In this talk we illustrate this approach, by giving, under additional con-ditions, a geometric characterization of flat groups of automorphisms; theseare analogues of tori in semisimple algebraic groups. For groups G whose

Short Communications 21

space B(G) is proper as a metric space, we show that a group H of auto-morphisms of G whose orbits in the space of compact, open subgroups arefinite-dimensional quasi-flats is virtually flat.


Canonical connections on k-symplectic manifolds under reduction

Adara-Monica Blaga – Universitatea de Vest Timisoara

The aim of this paper is to find conditions on the canonical connection on a k-symplectic manifold such that it should be preserved by performing Marsden-Weinstein reduction


Milnor-Wood inequalities for manifolds locally isometric toproducts of hyperbolic planes

Michelle Bucher-Karlsson – Kungliga Tekniska hogskolan Stockholm

Joint work with Tsachik Gelander

I will show how to use bounded cohomology, representation theory and su-perrigidity to obtain sharp Milnor-Wood inequalities for the Euler number offlat bundles over closed manifold locally isometric to products of hyperbolicplanes. It then follows that such manifolds cannot admit an affine structure,confirming Chern-Sullivan’s conjecture in this case. Existence and uniquenessof flat structures will also be discussed.


Degenerations of (pre)-Lie algebras

Dietrich Burde – Universitat Wien

We will discuss degenerations, contractions and deformations of Lie algebrasand pre-Lie algebras. While the Lie algebra case (and other cases, like Leibnizalgebras, or associative algebras) has been studied, the case of pre-Lie alge-bras still needs to be investigated. This class of algebras shows up in quantummechanics, i.e., in renormalization theory, and in geometry. Some resultsfrom the Lie case can be translated to the pre-Lie case. Among them are theresults concerning the cohomology of Lie algebras. Here we need a coho-mology theory for pre-Lie algebras. To obtain new semi-invariants for pre-Liealgebra degenerations, it is also interesting to consider generalized cocycles;for example generalized derivations, i.e., linear maps D : A → A satisfyingαD(x · y) = βD(x) · y + γx · D(y) for all x, y ∈ A and fixed α, β, γ ∈ K.Furthermore, there are new results for pre-Lie algebra degenerations which


have no analogue in the Lie algebra case.Scheduled on Monday afternoon.

Faithful representations of minimal dimension of currentHeisenberg Lie algebras

Leandro Cagliero – Universidad Nacional de Cordoba

Joint work with Nadina Rojas

Let g be a finite dimensional Lie algebra over a field k of characteristic zero k.By Ado’s Theorem we know that g admits a faithful representation. However,given an arbitrary Lie algebra g, the number

µ(g) = min{dimV : (π, V ) is faithful representation of g}

is, in general, very difficult to compute. Given a non-zero polynomial p ∈ k[t],the current Heisenberg Lie algebra associated to p is the Lie algebra

hm,p = hm ⊗ k[t]/(p)

where hm is the Heisenberg Lie algebra over k of dimension 2m + 1 and thebracket is given by [X1⊗ q1, X2⊗ q2] = [X1, X2]⊗ q1q2, Xi ∈ hm, qi ∈ k[t]/(p),i = 1, 2. This family of Lie algebras over k includes as a subfamily the classof truncated Heisenberg Lie algebras and the class of Heisenberg Lie algebrasover finite field extensions of k viewed as Lie algebras over k. In a recent jointwork with N. Rojas we obtained the following theorem.

Theorem. Let m ∈ N and p ∈ k[t], p 6= 0. Then

µ(hm,p) = mdeg p+⌈2√

deg p⌉.

This extends the known result µ(hm) = m + 2. In this talk we will brieflydescribe the proof of this theorem.


The automorphism group of accessible groups and of hyperbolicgroups

Mathieu Carette – Universite Libre de Bruxelles

Let G be a group acting on a tree. It has been shown in various contexts thatthis action not only clarifies the structure of the group itself, but also that ofits (outer) automorphism group. A classical example is that of a group actingcocompactly on a tree with trivial edge stabilizers, in which case G is a freeproduct of some groups Gi. In this context, Fouxe-Rabinovitch proved that iffor each i, Gi is freely indecomposable and both Gi and Aut(Gi) are finitely


presented, then Aut(G) is also finitely presented. Other examples include theJSJ decomposition of a one-ended (Gromov) hyperbolic group G, from whichSela and Levitt extracted structural results on Aut(G). Suppose that G actson a tree T with finite edge stabilizers. We give sufficient conditions on theaction of G on T for the automorphism group of G to be finitely presented. Inparticular, we show that the automorphism group of any (Gromov) hyperbolicgroup is finitely presented.


Groups of affine transformations generated by inversions

Virginie Charette – Universite de Sherbrooke

Joint work with Todd Drumm and William Goldman

Let E denote three-dimensional Minkowski space. Considering the hyperbolicplane H as a quadric inside E, reflections in H correspond to spacelike andtimelike inversions in E. We might ask, when does a group G generatedby these inversions act properly on E? For instance, following the work ofDrumm, we can associate a “crooked plane” to each generating inversion; ifthe crooked planes are pairwise disjoint, then G acts properly discontinuouslyon E. We will discuss the configuration space of pairs and triples of pairwisedisjoint crooked planes, and describe the corresponding space of groups. Wewill relate this to recent joint work with Drumm and Goldman, on the holon-omy of a three-holed sphere.


Simply transitive NIL-affine actions and LR-structures

Sandra Deschamps – Katholieke Universiteit Leuven Campus Kortrijk

Joint with Dietrich Burde and Karel Dekimpe

As a natural generalization of the usual affine group Aff(Rn) = RnoGL(n,R),we consider the affine group of a connected and simply connected nilpotentLie group N , which is defined as Aff(N) = N o Aut(N) and which acts on Nvia (m,α)n = m · α(n), for all m,n ∈ N , α ∈ Aut(N).

We will focus on simply transitive actions of one simply connected nilpo-tent Lie group G on another one, say N , via a map ρ : G → Aff(N) and referto such actions as NIL-affine actions.

In the usual affine case (i.e. N = Rn) the notion of a simply transitiveaffine action has been translated completely towards the Lie algebra level. Weshow that an analogous translation is available for the much more generalsimply transitive NIL-affine actions. This allows us to easily present examplesand counterexamples in low dimensions.


We then focus on abelian simply transitive NIL-affine actions, i.e. ρ : Rn →Aff(N), as a nice setting next to the affine case. Thereby we discovered thatthe existence of such an abelian NIL-affine action is equivalent to the existenceof a special Lie-compatible algebra structure, called LR-structure, on the Liealgebra of N .

We prove that a Lie algebra admitting an LR-structure, has to be two-step solvable. We prove the existence of an LR-structure on several classes oftwo-step solvable Lie algebras. Conversely we also present an example of athree-step nilpotent Lie algebra on four generators without an LR-structure.


Growth in Baumslag-Solitar groups: subgroups and rationality

Eric Freden – Southern Utah University

The computation of growth series for the higher Baumslag-Solitar groups isan open problem first posed by de la Harpe and Grigorchuck. We study thegrowth of the horocyclic subgroup as the key to the overall growth of theseBaumslag-Solitar groups BS(p, q), where 1 < p < q. In fact, the overall growthseries can be represented as a modified convolution product with one of thefactors being based on the series for the horocyclic subgroup. We exhibit twodistinct algorithms that compute the growth of the horocyclic subgroup anddiscuss the time and space complexity of these algorithms. We show thatwhen p divides q, the horocyclic subgroup has a geodesic combing whosewords form a context-free language. A theorem of Chomsky-Schutzenbergerallows us to compute the growth series for this subgroup, which turns out tobe rational. When p does not divide q, we show that no geodesic combingfor the horocyclic subgroup forms a context-free language, although there isa context-sensitive geodesic combing. We exhibit a specific linearly boundedTuring machine that accepts this language in the case of BS(2, 3). We will alsodemonstrate graphic software that illustrates this context-sensitive combing.


Global hyperbolicity of left-invariant Lorentz metrics on Liegroups

Mohammed Guediri – King Saud University

We show that every left-invariant Lorentz metric on a non-abelian simply con-nected Lie group is globally hyperbolic whenever its restriction to the derivedideal of the Lie algebra is positive definite. We then apply this result to thecase of 2-step nilpotent Lie groups. For instance, we show that a left-invariantLorentz metric on the three dimensional Heisenberg group is globally hyper-bolic if and only if its restriction to the center of the Lie algebra is positive


definite or degenerate.Scheduled on Tuesday afternoon.

Properties of Kahler flat manifolds

Marek Halenda – Uniwersytet Gdanski

Let M be a Kahler flat manifold. Then its fundamental group Γ is a discretecocompact torsion-free subgroup of U(n) n Cn. Equivalently, Γ is a Bieber-bach group of even dimension such the that associated holonomy represen-tation ϕ : H −→ GL(2n,R) has the following property: each R-irreduciblesummand of ϕ which is also C-irreducible occurs with even multiplicity. Us-ing this characterization, we will show examples of Kahler flat manifolds anddetermine some of their invariants and properties. We will be particularly in-terested in the class of complex Hantzsche-Wendt manifolds. These complexanalogues of well known Hantzsche-Wendt manifolds are defined as complexn-dimensional manifolds with holonomy H isomorphic to Zn−1

2 and containedin SU(n). We will describe that class in detail using a construction inspired bymethods from the theory of complex abelian varieties.


On some properties of submanifolds in golden Riemannianmanifolds

Cristina Hretcanu – Universitatea Stefan cel Mare

The main purpose of the present paper is to point out and find some applica-tions of the Golden Ratio and of Fibonacci numbers on Riemannian manifolds.We say that an (1,1)-tensor field P on an m-dimensional Riemannian mani-fold (M, g) is a Golden Structure if it satisfies the equation P 2 = P + Id(which is similar to that satisfied by the Golden Ratio φ) where Id stands foridentity. A Riemannian manifold endowed with a Golden Structure will becalled a Golden Riemannian manifold. We give an effective construction forsome induced structures on product of spheres of codimension r (r > 1) in anEuclidean space endowed with a Golden Structure.


Seifert fibered structure and rigidity on real Bott towers

Yoshinobu Kamishima – Tokyo Metropolitan University

Given an upper triangular matrix A whose diagonal entries are all 1 and off-diagonal entries are either 0 or 1, it assigns to A a smooth free (Z/2)n - actionon the n-dimensional torus Tn. The orbit space M(A) is an n-dimensional eu-clidean space form, called a real Bott tower. We shall prove that every real Bott


tower M(A) admits an injective Seifert fibred structure, that is there exists ak-torus action onM(A) whose quotient space is some (n−k)-dimensional realBott tower orbifold by some (Z/2)s-action, (1 ≤ s ≤ k.) Then by the Seifertrigidity, we can distinguish the diffeomorphism classes of real Bott towers andshow that all n-dimensional real Bott towers are obtained from the lower di-mensional real Bott towers, that is n− k-dimensional actions ((Z/2)s,M(B))(1 ≤ s ≤ k), k = 1, . . . , n. Originally a Bott tower is a sequence of complexprojective line-bundles defined by Grossberg-Karshon. A real Bott tower ischaracterized as a the set of real points of the Bott tower, that is a sequenceof real projective line-bundles, which is also a lagrangian submanifold.


Analogues of curve complex for free groups

Ilya Kapovich – University of Illinois At Urbana-Champaign

Joint work with Martin Lustig

We introduce several possible analogues of the curve complex for the freegroup FN , N ≥ 2 and study the action of Out(FN ) on them via Bonahon-type “intersection forms”. We prove that for N > 2 all these complexes haveinfinite diameter.


The geometry of Kleinian groups with real parameters

Elena Klimenko – Heinrich-Heine-Universitat Dusseldorf

Kleinian groups are the discrete subgroups of PSL(2,C), the full group of ori-entation preserving isometries of hyperbolic 3-space. The classification of hy-perbolic 3-manifolds was given earlier in topological terms together with theproof of Thurston’s ending lamination conjecture and Marden’s tameness con-jecture. However, there are still open problems in the area. For example, itis still unknown in general whether two matrices of SL(2,C) generate a dis-crete group, and if they do, what is the quotient orbifold. We concentrateon the two-generator Kleinian groups with real traces of the generators andtheir commutator, and give a complete classification of such groups and cor-responding orbifolds.


Counting growth types of automorphisms of free groups

Gilbert Levitt – Universite de Caen

Given an automorphism of a free group Fn, we consider the following invari-ants: e is the number of exponential strata (an upper bound for the number


of different exponential growth rates of conjugacy classes); d is the maximaldegree of polynomial growth of conjugacy classes; R is the rank of the fixedsubgroup. We determine precisely which triples (e, d,R) may be realized byan automorphism of Fn. In particular, the inequality e ≤ (3n − 2)/4 (due toLevitt-Lustig) always holds.


Volume entropy for hyperbolic buildings

Seonhee Lim – Cornell University, Ithaca

Joint work with Francois Ledrappier

Volume entropy of a Riemannian manifold is the exponential growth rate ofthe volumes of balls. Entropy rigidity for rank-1 Riemannian manifolds isknown: a theorem of Besson-Courtois-Gallot says that the locally symmetricmetrics attain minimal volume entropy among all Riemannian metrics. In thistalk, we are interested in entropy rigidity for buildings, especially hyperbolicones. We will give several characterizations of the volume entropy, analogousto the ones for trees, that will help us to find some lower bound on volumeentropy for certain hyperbolic buildings.


Trees versus currents, for automorphisms of free groups

Martin Lustig – Universite Paul Cezanne

Joint work with Ilya Kapovich

R-trees, currents, algebraic laminations and the rather delicate relationshipbetween them will be discussed. Some applications to automorphisms of freegroups will be pointed out. Particular attention will be given to perpendiculartree-current pairs.


On symmetry of flat manifolds

Rafal Lutowski – Uniwersytet Gdanski

Let X be a compact, connected, flat Riemannian manifold (flat manifold forshort) and let Γ be the fundamental group of X. Then Γ is a Bieberbachgroup, i.e. a torsion-free crystallographic group. Up to affine equivalence, Xis determined by Γ. Moreover if the first Betti number of X equals zero, thenthe group of affine self equivalences of X and the outer automorphism groupof Γ are isomorphic.

We consider a question: which finite groups occur as outer automorphismgroups of Bieberbach groups with trivial center? We present an example of


Γ, such that Out(Γ) is a cyclic group of order 3. We also calculate the outerautomorphism group of a direct product of n copies of Γ, for n ∈ N. Asa corollary we get a family of centerless Bieberbach groups Γn with Out(Γn)isomorphic to Sn, the symmetric group on n letters, for n ∈ N.


Classification of Exotic circles of Pr(S1)

Habib Marzougui – University of Bizerte

Joint work with Abdelhamid Adouani

In this paper, we deal with the existence of exotic circles in some groups ofcircle homeomorphisms: this means the existence of subgroups of a givengroup G that are topologically conjugate to the group SO(2) consisting ofcircle rotations but not conjugate to SO(2) in G. The existence of exoticcircle shows that the subgroup G is far from being a Lie group. We proveon one hand, that the group Pr(S1) (r ∈ N ∪ {+∞}, or r = ω) consistingof piecewise Cr class P circle homeomorphisms has no exotic circle, onthe other hand, its subgroup Pr1(S1) consisting of elements for which theproduct of all jumps at the break points is trivial has exotic circle, and we givea classification of them.


Conformal flows on compact pseudo-Riemannian manifolds

Karin Melnick – Yale University

Joint work with Charles Frances

By a celebrated theorem of Lelong-Ferrand and Obata affirming a conjectureof Lichnerowicz, a compact Riemannian manifold with noncompact conformalgroup is conformally equivalent to the round sphere Sn. Conformal dynamicson the type-(p, q) Einstein spaces, the pseudo-Riemannian analogues of theround sphere, are more complicated than the source-sink dynamics on Sn. Iplan to discuss the tight bound on the degree of a nilpotent group of conformalflows on any compact pseudo-Riemannian manifold M , and our theorem thatany M admitting a group of conformal automorphisms of maximal degree islocally conformally equivalent to flat Rp,q. These results support the pseudo-Riemannian Lichnerowicz conjecture.



Faithful representations of minimal degree for reductive Liealgebras

Wolfgang Moens – Universitat Wien

We consider Lie algebras of finite dimension over the complex numbers. Everysuch Lie algebra g allows a faithful, finite-dimensional, linear representationand we investigate the minimal degree of these representations:

µ(g) = min{ n | ∃ g ↪→ gln }.

For certain classes of Lie algebras, this µ-invariant can be obtained from clas-sical results in the theory of representations. The work of Schur (and alsoof Jacobson) for example, deals with the abelian case. The theory of highestweights gives us the µ for simple Lie algebras.

The calculation of the µ-invariant for semisimple Lie algebras is reducedto combinatorics by decomposing representations of semisimple Lie algebrasinto irreducible representations of the simple ideals. This also allows us tocharacterize these faithful representations of minimal degree.

Reductive Lie algebras are just the direct sums of abelian and semisimpleLie algebras. Representations of reductive Lie algebras are then split up intopairs of representations: one for the semisimple commutator and one for theabelian center. The calculation of the µ-invariant for reductive Lie algebrasis also reduced to combinatorics by explicitely calculating the centralizers ofrepresentations of semisimple Lie algebras.

In particular, we have a method to determine the reductive subalgebras ofgln, up to isomorphism. Similar problems were investigated by Mal’cev, Suter,Borel and Dynkin. They gave a complete classification of the (proper) maxi-mal reductive subalgebras of gln. Since every (proper) reductive subalgebra ofgln is contained in a (proper) maximal reductive subalgebra of gln, this classi-fication gives us a second way to calculate the µ-invariant for some reductiveLie algebras.


The Anosov theorem for infra-nilmanifolds with a 2-perfectholonomy group

Pieter Penninckx – Katholieke Universiteit Leuven Campus Kortrijk

Joint work with Karel Dekimpe and Bram De Rock

To a continuous map f : M → M on a closed manifold M , two numbersare assigned that are of particular interest in fixed point theory: the Lef-schetz number L(f) and the Nielsen number N(f). In this talk, we show thatN(f) = |L(f)| for any continuous map f : M → M on an infra-nilmanifoldM of which the holonomy group is 2-perfect (i.e. having no index two sub-group). Conversely, for any finite group F having an index two subgroup, we


show that there exists an infra-nilmanifold M with holonomy group F and acontinuous map f : M →M such that N(f) 6= |L(f)|.


Homology of Hantzsche-Wendt Groups

Nansen Petrosyan – Katholieke Universiteit Leuven Campus Kortrijk

Joint work with Karel Dekimpe

An n-dimensional Hantzsche-Wendt group is an n-dimensional orientable Bie-berbach group with holonomy group Z2

n−1. We develop an algorithm thatcomputes the homology of any Hantzsche-Wendt group by constructing apractical free resolution induced from the crystallographic action of the groupon Rn. As applications we compute the homology of all five and seven dimen-sional Hantzsche-Wendt groups.


Floyd Map and Convergence Groups

Leonid Potyagailo – Universite de Lille 1

Joint work with Victor Gerasimov

We study geometrically finite convergence groups of homeomorphisms of com-pact Hausdorff spaces. Let G be a convergence group of homeomorphisms ofa compact Hausdorff space T and ∂FG be the Floyd boundary of the Cayleygraph of G. It was previously shown by V. Gerasimov that if G acts cocom-pactly on the set Θ2(T ) of distinct pairs of T then there exists a G-equivariantcontinuous map F : ∂FG→ T . Our first result is the following :

Theorem 1. Let G be a convergence group of homeomorphisms of a compactumT acting cocompactly on the set Θ2(T ) of distinct pairs of T . Then for anyparabolic point p ∈ T the set F−1(p) is the Floyd boundary of the stabilizerHp = StabG(p) of p for the action GT .

Note that the the cocompactness on pairs by results of V. Gerasimov andP. Tukia is equivalent to the relative hyperbolicity for the action of G on T(i.e. every point of T is either conical or parabolic). In this content we willdescribe some applications of Theorem 1. In particular, we prove :

Theorem 2. Suppose that G is as above and H is a finitely generated groupadmitting a quasi-isometric embedding into G. Then H is relatively hyperbolicwith respect to parabolic subgroups situated in a bounded distance from those ofG: In particular parabolic subgroups of G are quasi-convex.


Some other applications concerning dynamical properties of relatively hy-perbolic groups will also be described.


Spin structures on flat 4-manifolds

Bartosz Putrycz – University of Gdansk

I will introduce briefly the idea of the spin structure on a differentiable ori-entable manifold and present some results for the existence of spin structureson general manifolds and especially on flat manifolds.

I will explain how the problem of the existence of spin structures on flatmanifolds is translated to an algebraic problem with the Bieberbach groups –the fundamental groups of flat manifolds.

Finally I will present the situation in dimension four and show some ex-amples of calculations.


Twinnings of R-buildings

Koen Struyve – Universiteit Gent

Joint work with Hendrik Van Maldeghem

We generalize the concept of affine twin buildings to the non-discrete caseof R-buildings. Using this new definition, these twin R-buildings are shownto have similar properties and structure as in the discrete case. In order todemonstrate the ideas behind the twinning of (R-)buildings, we highlight thecase of twin R-trees, which form the simplest examples of R-buildings. Weconclude with the construction of some examples of twin R-buildings.


LR-structures on nilpotent Lie algebras

Kim Vercammen – Katholieke Universiteit Leuven Campus Kortrijk

Joint work with Dietrich Burde and Karel Dekimpe

An LR-algebra is a vector space A equipped with a bilinear product · for whichboth the left and right multiplications commute, i.e.

(X · Y ) · Z = (X · Z) · Y and X · (Y · Z) = Y · (X · Z).

To each of these algebras we can associate a Lie algebra by defining the Lie-bracket by [X,Y ] = X · Y − Y ·X. We say that a Lie algebra g admits an LRstructure if there exists an LR product on g such that the associated Lie-bracketis the Lie-bracket on g. We say that an LR structure is complete if the right


multiplications are nilpotent. These complete LR-structures arise naturally inthe setting of NIL-affine actions (see abstract Sandra Deschamps).

In my talk I will discuss LR-structures on nilpotent Lie algebras. We willbe able to deduce several results, including the fact that each nilpotent Liealgebra that admits an LR-structure also admits a complete LR-structure.


List of Participants

Agueda Mate, Raquel. [email protected] (p. 19)Universidad de Castilla - La Mancha, Facultad de Derecho y Ciencias Sociales,Ronda de Toledo, s/n, 13071 Ciudad Real, Spain.Andersen, Jørgen. [email protected] (p. 11)Aarhus Universitet, Institut for Matematiske Fag, Ny Munkegade, Bygning1530, 8000 Aarhus C, Denmark.Appel, Daniel. [email protected] (p. 19)Heinrich-Heine-Universitat Dusseldorf, Mathematisches Institut, Universitats-strasse 1, 40225 Dusseldorf, Germany.

Barbot, Thierry. [email protected] (p. 20)Ecole Normale Superieure de Lyon, Unite de Mathematiques Pures et Ap-pliquees, UMR CNRS 5669, 46, allee d’Italie, 69364 Lyon , France.Baumgartner, Udo. [email protected] (p. 20)University of Newcastle, University Drive, Bd. V, Callaghan NSW 2308, Aus-tralia.Bienert, Rolf. [email protected] Heinrich-Heine Universitat Dussel-dorf, Onyxweg 8, D-41564 Kaarst, Germany.Blaga, Adara-Monica. [email protected] (p. 21)Universitatea de Vest Timisoara, Facultatea de Matematica si Informatica,Bulevardul Vasile Parvan nr. 4, Timisoara, Timis, 300223, Romania.Bourdon, Marc. [email protected] de Lille 1, UFR de Mathematiques, Cite Scientifique - Bat. M2,59655 Villeneuve d’Ascq , France.Bridson, Martin. [email protected] (p. 11)University of Oxford, Mathematical Institute, 24-29 St Giles’, Oxford, OX13LB, UK.Bucher-Karlsson, Michelle. [email protected] (p. 21)Kungliga Tekniska hogskolan Stockholm, Institutionen for Matematik, Lind-stedts vag 25, SE - 100 44 Stockholm, Sweden.Burde, Dietrich. [email protected] (p. 21)Universitat Wien, Fakultat fur Mathematik, Nordbergstrasse 15, 1090 Wien,

33

mailto:[email protected]












Austria.

Cagliero, Leandro. [email protected] (p. 22)Universidad Nacional de Cordoba, FAMAF-CONICET, National University ofCordoba, Ciudad Universitaria, 5000 Cordoba, Argentina.Caprace, Pierre-Emmanuel. [email protected] (p. 15)Institut des Hautes Etudes Scientifiques, Le Bois-Marie 35, Route de Chartres91440 Bures-sur-Yvette, France.Carette, Mathieu. [email protected] (p. 22)Universite Libre de Bruxelles, Departement de Mathematique, CP216, Boule-vard du Triomphe, 1050 Bruxelles, Belgium.Charette, Virginie. [email protected] (p. 23)Universite de Sherbrooke, Departement de Mathematiques, 2500 boul. del’Universite, Sherbrooke QC J1K 2R1, Canada.

De Clerck, Frank. [email protected] Gent, Vakgroep Zuivere wiskunde en computeralgebra, Krijgslaan281 S22, B-9000 Gent, Belgium.de la Harpe, Pierre. [email protected] (p. 15)Universite de Geneve, Section de Mathematiques, 2–4 rue du Lievre, CP 64,CH-1211 Geneve 4, Switzerland.De Medts, Tom. [email protected] Gent, Vakgroep Zuivere wiskunde en computeralgebra, Krijgslaan281 S22, 9000 Gent, Belgium.Dekimpe, Karel. [email protected] Universiteit Leuven Campus Kortrijk, Departement Wiskunde, E-tienne Sabbelaan 53, B-8500 Kortrijk, Belgium.Deprez, Steven. [email protected] Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200B,B-3001 Heverlee, Belgium.Deschamps, Sandra. [email protected] (p. 23)Katholieke Universiteit Leuven Campus Kortrijk, Departement Wiskunde, E-tienne Sabbelaan 53, B-8500 Kortrijk, Belgium.Dreesen, Dennis. [email protected] Universiteit Leuven Campus Kortrijk, Departement Wiskunde, E-tienne Sabbelaan 53, B-8500 Kortrijk, Belgium.Duchesne, Bruno. [email protected] de Geneve, Section de Mathematiques, 2–4 rue du Lievre, CP 64,1211 Geneve 4, Switzerland.Dwyer, Christopher. [email protected] University of New York, Department of Mathematical Sciences, Bing-hamton, New York 13902-6000, USA.

El Kacimi, Aziz. [email protected] de Valenciennes et du Hainaut-Cambresis, Laboratoire de Mathema-
















tiques et de leurs Applications, ISTV 2, Le Mont Houy, 59313 ValenciennesCedex 9, France.

Felix, Yves. [email protected] Catholique de Louvain, Departement de Mathematique, Bat. M. deHemptinne - Chemin du Cyclotron, 2 - 1348 Louvain-la-Neuve, Belgium.Falguieres, Sebastien. [email protected] Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200B,B-3001 Heverlee, Belgium.Farb, Benson. [email protected] (p. 12)University of Chicago, Department of Mathematics, 5734 University Avenue,Chicago IL 60637-1514, USA.Freden, Eric. [email protected] (p. 24)Southern Utah University, Mathematics Department, 351 West University Boul-evard, Cedar City, UT 84720, USA.

Garion, Shelly. [email protected] University of Jerusalem, Einstein Institute of Mathematics, EdmondJ. Safra Campus, Givat Ram, Jerusalem 91904, Israel.Goldman, William. [email protected] of Maryland, Department of Mathematics, College Park, MD 20742-4015, USA.Grigorchuk, Rostislav. [email protected] A&M University and Institut des Hautes Etudes Scientifiques, Mathe-matics Department,College Station, TX 77843-3368 , USA.Grunewald, Fritz. [email protected] (p. 12)Heinrich-Heine-Universitat Dusseldorf, Mathematisches Institut, Universitats-strasse 1, 40225 Dusseldorf, Germany.Guediri, Mohammed. [email protected] (p. 24)King Saud University, Department of Mathematics, College of Science, P.O.Box2455, Riyadh 11451, Saudi Arabia.

Halenda, Marek. [email protected] (p. 25)Uniwersytet Gdanski, Instytut Matematyki, Wita Stwosza 57, 80-952 Gdansk,Poland.Hamenstadt, Ursula. [email protected] (p. 12)Rheinische Friedrich-Wilhelms-Universitat Bonn, Mathematisches Institut, Be-ringstrasse 1, 53115 Bonn, Germany.Hartl, Manfred.Universite de Valenciennes et du Hainaut-Cambresis, Laboratoire de Mathema-tiques et de leurs Applications, ISTV 2, Le Mont Houy, 59313 ValenciennesCedex 9, France.Hretcanu, Cristina. [email protected] (p. 25)Universitatea Stefan cel Mare, Str. Universitatii, no.13, 720229 Suceava, Ro-mania.












mailto:



Igodt, Paul. [email protected] Universiteit Leuven Campus Kortrijk, Departement Wiskunde, E-tienne Sabbelaan 53, B-8500 Kortrijk, Belgium.

Ji, Lizhen. [email protected] (p. 15)University of Michigan, Department of Mathematics, 1834 East Hall, 530Church Street, Ann Arbor, MI 48109-1043, USA.

Kamishima, Yoshinobu. [email protected] (p. 25)Tokyo Metropolitan University, Department of Mathematics, 1 - 1 Minami-Ohsawa, Hachioji-shi, Tokyo, 192-0397, Japan.Kapovich, Ilya. [email protected] (p. 26)University of Illinois At Urbana-Champaign, Department of Mathematics, 1409West Green Street, Urbana, IL 61801, USA.Karmakar, Biswajit. [email protected] fur Mathematik in den Naturwissenschaften, Inselstrasse22-26, D-04103 Leipzig, Germany.Klimenko, Elena. [email protected] (p. 26)Heinrich-Heine-Universitat Dusseldorf, Mathematisches Institut, Universitats-strasse 1, 40225 Dusseldorf, Germany.Knebusch, Anselm. [email protected] and U. Gottingen, Departement Wiskunde, Celestijnenlaan 200B,B-3001 Heverlee, Belgium.

Levitt, Gilbert. [email protected] (p. 26)Universite de Caen, Boulevard du Marechal Juin 14032 Caen, France.Lim, Seonhee. [email protected] (p. 27)Cornell University, Ithaca, Department of Mathematics, 310 Malott Hall, Ithaca,NY 14853-4201, USA.Lubotzky, Alex. [email protected] (p. 13)Hebrew University of Jerusalem, Einstein Institute of Mathematics, EdmondJ. Safra Campus, Givat Ram, Jerusalem 91904, Israel.Lustig, Martin. [email protected] (p. 27)Universite Paul Cezanne, Departement de Mathematiques et LATP, BatimentHenri Poincare, Faculte des sciences et techniques, Avenue de l’escadrille Nor-mandie-Niemen, 13397 Marseille Cedex 20, France.Lutowski, Rafal. [email protected] (p. 27)Uniwersytet Gdanski, Instytut Matematyki, Wita Stwosza 57, 80-952 Gdansk,Poland.

Muhlherr, Bernhard. [email protected] (p. 16)Universite Libre de Bruxelles, Departement de Mathematique, CP216, Boule-vard du Triomphe, 1050 Bruxelles, Belgium.















Mallahi-Karai, Keivan. [email protected] University of Bremen, Mathematical Sciences, P.O. Box 750 561, 28725Bremen, Germany.Marquis, Ludovic. [email protected] Paris 6, 4 Place Jussieu, 75005 Paris, France.Marzougui, Habib. [email protected] (p. 28)University of Bizerte, Department of Mathematics, Faculty of Sciences of Biz-erte, 7021 Zarzouna, Tunisia.Melnick, Karin. [email protected] (p. 28)Yale University, Department of Mathematics, PO Box 208283, New Haven, CT06520-8283, USA.Melotte, Tom. [email protected] Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200B,B-3001 Heverlee, Belgium.Moens, Wolfgang. [email protected] (p. 29)Universitat Wien, Fakultat fur Mathematik, Nordbergstrasse 15, 1090 Wien,Austria.Monod, Nicolas. [email protected] (p. 13)Ecole Polytechnique Federale de Lausanne, SB IMB EGG, MA B1 447 (BatimentMA), Station 8 , CH-1015 Lausanne, Switzerland.Moon, Soyoung. [email protected] de Neuchatel, Institut de Mathematiques, Rue Emile-Argand 112007 Neuchatel, Switzerland.

Palesi, Frederic. [email protected] Fourier, 100 rue des Maths, BP 74, 38402 St Martin d’Heres, France.Parker, John. [email protected] (p. 13)University of Durham, Department of Mathematical Sciences, Science Labora-tories, South Road, Durham, DH1 3LE, Uk.Penninckx, Pieter. [email protected] (p. 29)Katholieke Universiteit Leuven Campus Kortrijk, Departement Wiskunde, E-tienne Sabbelaan 53, B-8500 Kortrijk, Belgium.Petrosyan, Nansen. [email protected] (p. 30)Katholieke Universiteit Leuven Campus Kortrijk, Departement Wiskunde, E-tienne Sabbelaan 53, B-8500 Kortrijk, Belgium.Potyagailo, Leonid. [email protected] (p. 30)Universite de Lille 1, UFR de Mathematiques, Cite Scientifique - Bat. M2,59655 Villeneuve d’Ascq , France.Putrycz, Bartosz. [email protected] (p. 31)University of Gdansk, Instytut Matematyki,ul. Wita Stwosza 57,80-952 Gdansk,Poland.

Ribnere, Evija. [email protected] Dusseldorf, Mathematisches Institut, Universitats-

















strasse 1, 40225 Dusseldorf, Germany.

Sabbe, Marie. [email protected] Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200B,B-3001 Heverlee, Belgium.Struyve, Koen. [email protected] (p. 31)Universiteit Gent, Vakgroep Zuivere wiskunde en computeralgebra, Krijgslaan281 S22, 9000 Gent, Belgium.Sulway, Robert. [email protected] of California, Department of Mathematics, South Hall, Room 6607,Santa Barbara, CA 93106, USA.

Vaes, Stefaan. [email protected] (p. 17)Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200B,B-3001 Heverlee, Belgium.Valette, Alain. [email protected] de Neuchatel, Institut de Mathematiques, Rue Emile-Argand 11,2007 Neuchatel, Switzerland.Vander Vennet, Nikolas. [email protected] Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200B,B-3001 Heverlee, Belgium.Vercammen, Kim. [email protected] (p. 31)Katholieke Universiteit Leuven Campus Kortrijk, Departement Wiskunde, E-tienne Sabbelaan 53, B-8500 Kortrijk, Belgium.Verheyen, Kelly. [email protected] Universiteit Leuven Campus Kortrijk, Departement Wiskunde, E-tienne Sabbelaan 53, B-8500 Kortrijk, Belgium.Vogtmann, Karen. [email protected] (p. 17)Cornell University, Ithaca, Department of Mathematics, 310 Malott Hall, Ithaca,NY 14853-4201, USA.

Wortman, Kevin. [email protected] of Utah, Department of Mathematics, 155 South 1400 East, Room233, Salt Lake City, UT 84112-0090, USA.Wouters, Tim. [email protected] Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200B,B-3001 Heverlee, Belgium.













with gratitude to

• the Fund for Scientific Research – Flanders,

• the Fonds National de la Recherche Scientifique,

• the F.W.O. Research Network WO.003.01N, Fundamen-tal methods and Techniques in Mathematics,

• the Universite Catholique de Louvain (UCL),

• the Katholieke Universiteit Leuven (Campus Kortrijk)

for their support.

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