Discrete groups and Geometric Structures with Applications · Discrete Groups and Geometric...

Conference Booklet of the 4th Kortrijk Workshop

Discrete groups and Geometric Structureswith Applications

Oostende, May 31 - June 3, 2005

Karel Dekimpe, Paul Igodt and Hannes Pouseeleorganizers

Yves Felix, William Goldman, Fritz Grunewald,Paul Igodt and Kyung Bai Lee

scientific committee

Conference Booklet 3

Dear participant,

we wish you a very warm welcome on this fourth edition of the Kortijk workshop onDiscrete Groups and Geometric Structures, with Applications.

After three editions that took place at our home institution, the campus Kortrijk of the

Katholieke Universiteit Leuven, we decided to rely on the hospitality of the Hotel Royal

Astrid and the agreeable atmosphere of the Belgian coast as setting for this fourth edition.

We hope you will enjoy this conference, its lectures and discussions both on a mathe-matical and a personal level.

Karel Dekimpe, Paul Igodt and Hannes Pouseele

Schedule 5

ScheduleInvited speakers are Oliver Baues (Karlsruhe), Yves Benoist (Paris), Martin Bridson

(London), Benson Farb (Chicago), Oscar Garcia-Prada (Madrid), Etienne Ghys (Lyon)and Domingo Toledo (Utah). Each of them is giving a one-hour talk. Next to the parallelsessions and the poster session, there are also 4 plenary sessions of 45 minutes each. Allplenary sessions take place in the auditorium. The parallel sessions take place in the roomsIridia 1 (IR1) and Iridia 3 (IR3).

Tuesday, May 31

Morning session 1

08.30-09.30 Registration

09.30-09.45 Welcome and Opening of the Workshop by P. Igodt.

09.45-10.45 E. Ghys: Quasi-morphisms on SL(2, Z): old and new

10.45-11.15 Coffee break

11.15-12.15 O. Baues: Homotopy equivalences of solv-manifolds and

arithmeticity

Afternoon session 1

14.00-14.25IR1 D. Osajda: Boundaries of systolic groups

IR3 K. Dekimpe: The Auslander conjecture for NIL-affine

crystallographic groups14.35-15.00

IR1 T. Barbot: On the dynamics of surface groups on the flag

varietyIR3 I. Kim: On Marcus conjecture

15.10-15.35IR1 E. Breuillard: Uniform versions of the Tits alternative

IR3 M. Guediri: A new class of compact spacetimes without

closed nonspacelike geodesics


16.10-16.35IR1 H. Abels: A question of C.L. Siegel

IR3 A. Thomas: Covolumes of Uniform Lattices acting on Hyper-

bolic Buildings

16.45-17.30 G. Link: Geometry of discrete subgroups of higher rank Lie

groups

6 Discrete Groups and Geometric Structures II

Wednesday, June 1

Morning session 2

09.00-10.00 D. Toledo: Non-arithmetic uniformizations of some real moduli

spaces


10.30-11.30 Y. Benoist: 3-dimensional projective tilings

11.40-12.05IR1 K. Altmann: Hyperbolic lines in unitary space

IR3 S. Choi: Projective structures on 3-dimensional orbifolds and

some deformations

Afternoon session 2

14.00-14.25IR1 J. Lauret: A canonical compatible metric for geometric struc-

tures on nilmanifoldsIR3 Y. de Cornulier: Finitely presented wreath products

14.35-15.00IR1 D. Burde: Classical Yang-Baxter Equation and Geometry of Lie

GroupsIR3 N. Cotfas: Permutation representations defined by G-clusters

with application to quasicrystal physics

15.10-16.00 Poster Session


16.30-17.15 H. Pouseele: Betti number behavior for nilpotent Lie algebras

Schedule 7

Thursday, June 2

Morning session 3

09.00-10.00 B. Farb: Problems and progress in understanding the

Torelli group


10.30-10.55IR1 M. Wolff: Non-injective representations of a closed surface

group into PSL(2, R)IR3 Y. Kamishima: Cusp cross-sections of hyperbolic orbifolds by

Heisenberg nilmanifolds11.05-11.30

IR1 P. Dani: Finding the density of finite order elements in infinite

groupsIR3 B. McReynolds: Arithmetic cusp shapes are dense

11.40-12.25 I. Mineyev: Cohomology and hyperbolicity

Afternoon Social activity / Excursion

Evening Conference Dinner


Friday, June 3

Morning session 4

09.00-10.00 O. Garcia–Prada: G-Higgs bundles and surface group

representations


10.30-11.30 M. Bridson: Limit groups: non-positive curvature, group theory

and logic11.40-12.05

IR1 V. Charette: Affine deformations of the holonomy group

of a three-holed sphereIR3 A. Szczepanski: Endomorphisms of relatively hyperbolic groups

Afternoon session 3

14.00-14.25IR1 K. Melnick: Isometric actions of Heisenberg groups on compact

Lorentz manifoldsIR3 O. Talelli: On a theorem of Kropholler and Mislin

14.35-15.00IR1 K.B. Lee: Riemannian foliations on H

2 × R

IR3 P. Tumarkin: Hyperbolic Coxeter groups


15.40-16.25 W. Goldman: Proper affine actions and geodesic flows of

hyperbolic surfaces

16.35-16.45 Closing of the workshop

Abstracts of invited talks 9

Invited talks

Homotopy equivalences of solv-manifolds and arithmeticityOliver Baues Universitat Karlsruhe

To the fundamental group of every solv-manifold M and, more generally, to every torsion-free polycyclic by finite group we associate a linear algebraic group which is defined overthe rational numbers. This construction is functorial and determines many geometricproperties of the K(π, 1)-space M . In this talk, we explain that the group of homotopyclasses of self-equivalences of M is isomorphic to an arithmetic group.

3-dimensional projective tilingsYves Benoist ENS, Paris

We will study the following setting: Ω is a convex open subset of the real projective 3-spacewith a cocompact discrete group of projective transformations Γ. For instance, we will seethat, when Ω is indecomposable and Γ torsion free, the properly embedded flat trianglesin Ω project in the quotient M := Γ\Ω onto finitely many disjoint tori and Klein bottleswhich induce an atoroidal decomposition of M .

Limit groups: non-positive curvature, group theory and logicMartin Bridson Imperial College, London

In his solution to the Tarski problem, Zlil Sela described the class of “limit groups”. Limitgroups have Cayley graphs that arise as pointed Gromov-Hausdorff limits of Cayley graphsof free groups. Such groups L are precisely those with the property that, given any finitesubset S ⊂ L, there is a homomorphism to a free group that is injective on S. Theycan also be characterised by their first order logic, and in terms of the structure of theirclassifying spaces.

In this talk I shall outline the basic structure of limit groups (following Sela), describethe geometry of their classifying spaces, and then present recent work by Howie and I onthe subgroup structure of limit groups and, more strikingly, their direct products.


Problems and progress in understanding the Torelli groupBenson Farb University of Chicago

The Torelli group T (S) is defined to be the subgroup of the mapping class group con-sisting of those mapping classes acting trivially on H1(S, Z). The study of T (S) connectsto 3-manifold theory, symplectic representation theory, combinatorial group theory, andalgebraic geometry. In this talk I will describe some of the main themes in this beautifultopic, concentrating on the combinatorial and (co)homological aspects. I will also describesome recent progress in this direction, as well as a number of open questions and conjec-tures. Different parts of the talk will describe joint work with (separately) Daniel Biss,Nick Ivanov and Tara Brendle.

G-Higgs bundles and surface group representationsOscar Garcia-Prada CSIC, Madrid

In this talk we describe the theory of G-Higgs bundles over a compact Riemann surfacewhere G is a non-compact real reductive Lie group. We then show how this theory is usedto study the moduli space of representations of the fundamental group of the surface inG. Special emphasis is given to the case in which G/H is a Hermitian symmetric space,where H is a maximal compact subgroup of G.

Quasi-morphisms on SL(2, Z): old and newEtienne Ghys ENS, Lyon

A quasi-morphism on a group G is a map f from G to R which is a “morphism up to abounded error”, ie such that the modulus of f(xy)− f(x)− f(y) is uniformly bounded. Inthe first part of my talk, I plan to explain why these objects are worth studying in relationwith dynamics and topology. Then I will consider the case where G is the modular groupSL(2, Z) and give many examples. Some of these examples are very old but some othersare pretty recent.

Non-arithmetic uniformizations of some real moduli spacesDomingo Toledo IHES, Paris

This lecture will describe joint work with Allcock and Carlson on the structure of the realpoints of some complex moduli spaces. If the complex moduli space is complex hyperbolic,one expects the corresponding real moduli space to be real hyperbolic. This is correct, butnot necessarily in an obvious way. Namely, there may not be a single anti-holomorphicinvolution of complex hyperbolic space whose fixed point set uniformizes the real moduli

Abstracts of invited talks 11

space. This will be illustrated with the example of polynomials of degree 6, where thecomplex moduli space is known to be uniformized by complex hyperbolic 3-space. Themoduli space of real polynomials turns out to be real hyperbolic, containing 4 pieces thatparametrize the spaces of real polynomials with each of the 4 different possible configura-tions of real and complex conjugate pairs of roots. While each of the pieces is uniformizedby an arithmetic group, the group of the whole space is non-arithmetic, in the spirit ofa construction of Gromov and Piatetski-Shapiro. A more involved example of the samephenomenon is the moduli space of real cubic surfaces.


Plenary talks

Proper affine actions and geodesic flows of hyperbolic surfacesWilliam Goldman University of Marylandjoint work with F. Labourie and G. Margulis

Let F be a Schottky group inside SO(2, 1), and let S = H2/F be the correspondinghyperbolic surface. Let GC denote the space of geodesic currents on S. The cohomologygroup H1(F, V ) parametrizes equivalence classes of affine deformations of F acting on anirreducible representation V of SO(2, 1). We define a continuous biaffine map

M : GC × H1(F, V ) → R

which is linear on the vector space H1(F, V ). An affine deformation F[u] correspondingto a cohomology class [u] in H1(F, V ) acts properly if and only if M(c, [u]) is nonzero forall geodesic currents c. Consequently the set of proper affine actions whose linear partis a Schottky group identifies with a bundle of open convex cones in H1(F, V ) over theTeichmueller space of S.

Geometry of discrete subgroups of higher rank Lie groupsGabriele Link Universitat Karlsruhe

Let X = G/K be a higher rank symmetric space of noncompact type, ∂X its geometricboundary, and Γ ⊂ G a discrete group. In this talk we are going to investigate the structureof the geometric limit LΓ := Γx ∩ ∂X in purely geometrical terms. We will describe thedynamics of axial isometries which, together with an approximation argument originallydue to P. Eberlein ([4, Proposition 4.5.14]), leads to the main result

Theorem If Γ ⊂ G is “nonelementary”, then the regular geometric limit set splits

as a product KΓ × PΓ, where KΓ denotes the limit set in the Furstenberg boundary of X,

and PΓ is the set of Cartan projections of limit points.

Furthermore, KΓ is a minimal closed set under the action of Γ, and PΓ corresponds to

the closure of the set of translation vectors of axial isometries in Γ.

Although this result is already known for the smaller class of Zariski dense discretegroups (see i.e. [1], [2]), the advantage of our proof is its purely geometric nature whichallows to easily adapt the methods to products of pinched Hadamard manifolds (compare[3]). Furthermore, we obtain more insight into the action of individual isometries on thegeometric boundary. As an application, we describe a new construction of free groups inhigher rank symmetric spaces.

Abstracts of plenary talks 13

References.

[1] Y. Benoist, Proprietes asymptotiques des groupes lineaires I, Geom. Funct. Anal. 7 (1997), 1-47.

[2] J. P. Conze, Y. Guivarc’h, Limit Sets of Groups of Linear Transformations, Ergodic Theory and Harmonic Analysis,

Sankhya Ser. A 62 (2000), no. 3, 367-385.

[3] F. Dal’bo, I. Kim, Ergodic Geometry on the product of Hadamard manifolds, Preprint (2002).

[4] P. Eberlein, Geometry of Non-Positively Curved Manifolds, Chicago Lectures in Mathematics, Chicago Univ. Press,

Chicago, 1996.

[5] G. Link, Limit Sets of Discrete Groups acting on Symmetric Spaces,

www.ubka.uni-karlsruhe.de/cgi-bin/psview?document=2002/mathematik/9, Dissertation, Karlsruhe, 2002.

Cohomology and hyperbolicityIgor Mineyev University of Illinois

I will give a review of cohomological properties of hyperbolic groups, in particular the Ger-sten’s characterization of hyperbolic groups by the vanishing of the second `∞ cohomologywith `∞ coefficients. We will discuss recent developments in this area and new methodsused: a characterization of hyperbolicity by `∞ cohomology with various other coefficientsand cohomological properties of relatively hyperbolic groups.

Betti number behaviour for nilpotent Lie algebrasHannes Pouseele K.U.Leuven Campus Kortrijk

In general, it is quite a feat to determine the Betti numbers of all Lie algebras of a certaintype. One relies on deep combinatorial arguments to figure out a formula for, for instance,the Betti numbers of a Lie algebra containing an abelian ideal of codimension one. Incontrast to this combinatorial approach, I use arguments of spectral sequence type todetermine an explicit formula for the Betti numbers of low-dimensional split extensions ofa Heisenberg Lie algebra. Amongst others, the family of the twisted standard filiform Liealgebras is contained in this class.

Numerous examples, including all algebras with an abelian ideal of codimension 1, showa unimodal behavior, that is, βp < βp+1 for all p less or equal to half the dimension of thealgebra. This behavior is by no means a general phenomenon. We (joint work with DietrichBurde) use the explicit formulas developed above to show that the class of one-dimensionalextensions of a Heisenberg Lie algebra already contains examples, indeed whole families ofexamples, with a non-unimodal Betti number distribution.

The same kind of spectral sequence argument turns out to be useful in the context of theToral Rank Conjecture, stating that the total dimension of the cohomology of a nilpotentLie algebra is bounded below by 2z, where z is the dimension of the center. For splitmetabelian Lie algebras we (joint work with Paulo Tirao) construct an explicit embeddingof the cohomology of the center into the cohomology of the algebra, thus proving the ToralRank Conjecture for this class of algebras. This class of split metabelian algebras alreadycontains a family that doesn’t fit into the classical – combinatorial – approach of the ToralRank Conjecture.


Short Communications

A question of C.L. SiegelHerbert Abels Universitat Bielefeld

Let Γ = SL(n, Z) and G = SL(n, R). The main result of reduction theory describes afundamental set S for Γ in G, i.e., a subset S of G such that the natural map G → Γ\Grestricted to S, let us call it π : S → Γ\G, is surjective and has finite fibers. The setS described in reduction theory is called a Siegel set. Siegel asked in 1959 if Γ has theadditional property of preserving the natural metric up to a constant. In recent workwith G.A. Margulis we gave a positive answer to this question. Our solution has threeinteresting features:

1. It works for all reductive groups G over local fields and their corresponding arithmeticsubgroups. Positive answers to Siegels question had been known before for reductivegroups over the reals and their arithmetic subgroups (Ding, Ji, Leuzinger).

2. It works for all normlike metrics, not only for the metric coming from the symmetricspace (as considered by the authors above) or from the Bruhat-Tits building, butalso for word metrics.

3. The proof gives additional information in reduction theory.

Hyperbolic lines in unitary spaceKristina Altmann Technische Universitat Darmstadt

joint work with R. Gramlich

A central problem in synthetic geometry is the characterisation of graphs and geometries.The local recognition of locally homogeneous graphs forms one category of such character-isations. The game is the following. Choose a graph ∆, and try to identify all connectedgraphs which are locally ∆.

Let F be a finite field, F 6= F2, and K be a quadratic extension. We focus on then-dimensional vector space Vn over K endowed with a non-degenerate hermitian form anddefine the graph S(Vn) on the hyperbolic lines of Vn where two hyperbolic lines l and mare adjacent if and only if l is perpendicular to m with respect to the hermitian form.

For n ≥ 7 we show that any connected graph which is locally S(Vn) is isomorphic toS(Vn+2).

To obtain that result we first study graphs isomorphic to S(Vn) in order to recover thegeometry of Vn from the graph S(Vn). Next we prove that for n ≥ 8 the diameter of thegraph Γ is 2 and for n = 7 the diameter of the graph Γ is 3. This global property together

Abstracts of short communications 15

with the local properties we have studied before enables us to determine the isomorphismtype of Γ. As a corollary this leads to a characterisation of the group SUn+2(K), n ≥ 7,see Theorem 27.1 of [1].

Note that in the theorem the lower bound on n is in a sense optimal. Indeed, besides thegraph S(V8) also the graph on the fundamental SL2’s of the group 2E6(K) with commutingas adjacency is connected and locally S(V6), see Proposition 7.18 in Chapter 3 of [3] .

References

[1] Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The Classification of the Finite Simple Groups. American

Mathematical Society, 1994.

[2] Ralf Gramlich, On Graphs, Geometries and Groups of Lie Type. Eindhoven University Press, 2002.

[3] Franz Georg Timmesfeld, Abstract Root Subgroups and Simple Groups of Lie-Type. Birkh”auser, 2001.

On the dynamics of surface groups on the flag varietyThierry Barbot ENS, Lyon

Let Γ be the fundamental group of a closed surface with genus g > 1. Any morphism of Γin SL(2, R) induces a morphism ρ0 : Γ → G, where G ≈ SL(3, R) is the group of projectivetransformations of the projective plane. The induced projective action of Γ preserves apoint x0 and a projective line d0. The flag variety X is the set of pairs (x, d) (a flag) wherex is a point of the projective plane, and d a projective line containing x. The differential ofany projective Γ-action defines a Γ-action on X. For the projective action defined above,the associated Γ-action on X preserves a topological circle L0: the set of flags (x, d) wherex belongs to d0 and d contains x0.

A fundamental observation is that ρ0 is a (G, Y )-Anosov representation, in the meaningof F. Labourie, where Y is the space of triples of distinct points in RP 2. It follows thatfor morphisms ρ : Γ → G near to ρ0, the induced action on X still preserves a topologicalcircle L(ρ) in X, which, in general, is only Holder continuous. More precisely, if L(ρ) isLipschitz continuous, then its is analytic, and ρ(Γ) preserves a point or a projective linein the projective plane. Moreover, we obtain a satisfactory picture of the action of ρ(Γ)on X, showing that many common properties of the ρ0(Γ)-action remain valid, except oneimportant feature: there are two invariant annuli, and the dynamic on these invariantannuli highly depends on the morphism ρ. As a corollary, we can show that the action ofρ(Γ) on X is topologically conjugate to the action of ρ(Γ0) if and only if the representationsρ and ρ0 are conjugate inside G.

Uniform versions of the Tits alternativeEmmanuel Breuillard IHES, Paris

The celebrated Tits alternative states that a finitely generated non-virtually solvable lineargroup Γ contains a non-abelian free subgroup. We show that, for any finite generating setS of Γ, generators of such a free subgroup can be found as words of length at most r in


S, where r = r(Γ) is a constant depending on Γ only. This implies uniform exponentialgrowth as well as uniform non-amenability for finitely generated non virtually solvablelinear groups in arbitrary characteristic and thus subsumes the recent work of Eskin-Mozes-Oh. Although the constant r cannot be made entirely independent of Γ, we showthat it is uniform among all discrete subgroups of SLd(k), for an arbitrary local field k,where the dimension d is fixed, thus providing a lower bound for the algebraic entropyof discrete subgroups. Analogous results are obtained for the class of finitely generatedsolvable groups.

Classical Yang-Baxter Equation and Geometry of Lie GroupsDietrich Burde Universitat Wien

Let G be a Lie group with Lie algebra g. Any skewsymmetric element r ∈ Λ2(g) satisfyingthe CYBE

c(r) = [r12, r13] + [r12, r23] + [r13, r23] = 0

is called a classical r-matrix. Denote by T ∗G = g∗o G the cotangent bundle considered

as a Lie group where G acts on g∗ via the coadjoint action. Then there is a one-to-onecorrespondance between the r-matrices in G and the set of connected Lie subgroups of T ∗Gwhich carry a left-invariant affine structure and whose Lie algebras are Lagrangian graphsin g⊕ g∗. This is proved in [1]. We study a natural generalization of the notion of classicalr-matrices by replacing the adjoint action involved by an arbitrary g-module action. Thenone obtains left-invariant affine structures on Lie groups in much greater generality. Ourresults can also be applied to Novikov structures on Lie groups.

References.

[1] A. Diatta; A. Medina: Classical Yang-Baxter equation and left-invariant affine geometry on Lie groups. Manuscr.

Math. 114, No.4 (2004), 477-486.

Affine deformations of the holonomy group of a three-holedsphere

Virginie Charette University of Manitobajoint work with T.A. Drumm and W.M. Goldman

Suppose Γ = 〈γ1, γ2〉 is a free group of Lorentzian transformations. For each element of Γ,the linear part is either hyperbolic or parabolic.

In the hyperbolic case, a signed Lorenztian length of the associated closed geodesic α(γ)can be defined, and the sign of the transformation is the sign of the Lorentzian length of theassociated closed geodesic. This function can be extended to parabolic transformations. Iftwo transformations are of opposite sign then no group containing both of them can actproperly on E.


Suppose γ1 and γ2 are both hyperbolic and consider the invariant lines of their linearparts in the hyperbolic plane. When these lines are ultraparallel, Jones showed that ifthe sign of the generators and the sign of their product are the same then the group actsproperly discontinuously on E. When the lines intersect, checking the sign of any finitenumber of elements in the group will be inconclusive.

We will discuss these results and how they may be extended to parabolic generators.We will also describe the shape of the region of properly discontinuous actions in the spaceof Γ.

Projective structures on 3-dimensional orbifolds and somedeformations

Suhyoung Choi KAIST

We discuss some examples of 3-dimensional polyhedral orbifolds with projective structuresand their deformations. The techniques are from the higher dimensional Kac-Vinbergconstruction taken to the lower dimensions by taking 3-dimensional sections of the n-dimensional simplices.

Permutation representations defined by G-clusters withapplication to quasicrystal physics

Nicolae Cotfas University of Bucharest

Quasicrystals are systems with long-range order but non-periodic. Atomic structure de-termination of quasicrystals is a crucial issue in the understanding of physical properties.The diffraction pattern corresponding to a quasicrystal contains a set of bright spots in-variant under a finite group G, and the high-resolution electron microscopy suggests thatthe quasicrystal can be regarded as a quasiperiodic packing of interpenetrated copies of awell-defined G-invariant atomic cluster.

From a mathematical point of view, the G-cluster C describing the microstructure ofa quasicrystal is a finite union of orbits of G. By starting from the action of G on Cwe can define a permutation representation of G in a higher-dimensional Euclidean spaceEk = (Rk, 〈·, ·〉). The physical space can be identified with a G-invariant subspace E ⊂ Ek

in a canonical way. Using the strip projection method [1,2] we define the set

Q = πx | x ∈ Zk ∩ S

where π is the orthogonal projector corresponding to E and S is the strip [0, 1]k + Eobtained by translating the hypercube [0, 1]k along E. The set Q is a quasiperiodic packingof interpenetrating copies of C and can be regarded as a mathematical model for theconsidered quasicrystal [3-8].


References.

[1] A. Katz and M. Duneau, J. Phys. (France), 47(1986), 181.

[2] V. Elser, Acta Cryst. A, 42 (1986), 36.

[3] N. Cotfas and J.-L. Verger-Gaugry, J. Phys. A: Math. Gen., 30(1997), 4283.

[4] N. Cotfas, Lett. math. Phys., 47(1999), 111.

[5] N. Cotfas, J. Phys. A: Math. Gen., 32(1999), 8079.

[6] N. Cotfas, J. Phys. A: Math. Gen., 37(2004), 3125.

[7] N. Cotfas, Ferroelectrics, 305(2004), 33.

[8] http://fpcm5.fizica.unibuc.ro/~ncotfas

Finding the density of finite order elements in infinite groupsPallavi Dani University of Chicago

Consider a finitely generated infinite group Γ. Let P be a property that elements of Γmight have, such as having finite order, being contained in a maximal abelian subgroup orbeing self-centralizing. What is the probability that a given element of Γ has the propertyP? Fix a finite generating set S for Γ and let B(r) and E(r) denote the ball of radiusr and the set of elements with property P in the ball of radius r respectively. Define

FP (Γ, S) = limr→∞

|E(r)|

|B(r)|, if it exists. FP (Γ, S) measures the density of elements in Γ with

the property P . We consider the case when P is the property of having finite order andcompute this limit for a number of examples, including virtually nilpotent groups and wordhyperbolic groups, in which very different phenomena occur.

Finitely presented wreath productsYves de Cornulier EPF Lausanne

Proposition Let N 6= 1 and G be finitely presented groups, and X a G-set. Then thewreath product W = N (X)

o G is finitely presented if and only if(i) For every x ∈ X, the stabilizer Gx is finitely generated, and(ii) G has finitely many orbits on X2 (for the diagonal action).

This leads to the question: for what finitely generated groups G does there exist aninfinite G-set X satisfying (i)-(ii). We present some examples and some obstructions.


The Auslander conjecture for NIL-affine crystallographicgroups

Karel Dekimpe K.U.Leuven Campus KortrijkJoint work with D. Burde and S. Deschamps

Let N be a simply connected, connected real nilpotent Lie group of finite dimension n.We study subgroups Γ in Aff(N) = No Aut(N) acting properly discontinuously and co-compactly on N . This situation is a natural generalization of the so-called affine crystal-lographic groups. We prove that for all dimensions 1 ≤ n ≤ 5 the generalized Auslanderconjecture holds, i.e., that such subgroups are virtually polycyclic.

A new class of compact spacetimes without closednonspacelike geodesics

Mohammed Guediri KSU, Riyadh

Using simply transitive affine actions of the three-dimensional Heisenberg group H3 on R3,

we construct the first examples of geodesically complete compact spacetimes with regularglobally hyperbolic coverings but without closed nonspacelike geodesics.

Cusp cross-sections of hyperbolic orbifolds by Heisenbergnilmanifolds

Yoshinobu Kamishima Tokyo Metropolitan University

We study the geometric boundary problem. Long and Reid have shown that some compactflat 3-manifold cannot be diffeomorphic to a cusp cross-section of a 1-cusped finite volumehyperbolic manifold. Similar to the flat case, we give a negative answer that there exists a 3-dimensional closed Heisenberg infranilmanifold with cyclic holonomy of order bigger than orequal to 3, which cannot be diffeomorphic to a cusp cross-section of a 1-cusped finite volumecomplex hyperbolic 2-manifold.This is obtained from the formula by the characteristicnumbers of bounded domains related to the Burns-Epstein invariant on strictly pseudo-convex CR-manifolds. McReynolds informed us that Neumann-Reid have obtained thesimilar answer using eta-invariant. D. B. McReynolds gave a necessary and sufficientcondition for a Heisenberg infranilmanifold to be realized as a cusp-section of finite volume(arithmetically) complex hyperbolic orbifold (Algebr. and Geom. Topol. 2004 (4), 721-755). We study this problem using the Seifert fibration and group extensions of Heisenberginfranilmanifold.


On Marcus conjectureInkang Kim Seoul National University

joint work with K. Jo

We show that a compact affine manifold cannot have a parallel volume if the limit set ofthe identity component of the automorphism group of the universal cover of M , which isnot R

n, is nonempty.

A canonical compatible metric for geometric structures onnilmanifolds

Jorge Lauret FAMAF, Cordoba

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf.symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a leftinvariant Riemannian metric on N compatible with γ to be minimal, if it minimizes thenorm of the invariant part of the Ricci tensor among all compatible metrics with the samescalar curvature. We prove that minimal metrics (if any) are unique up to isometry andscaling, they develop soliton solutions for the ‘invariant Ricci’ flow and are characterized asthe critical points of a natural variational problem. The uniqueness allows us to distinguishtwo geometric structures with Riemannian data, giving rise to a great deal of invariants.If one considers no structure (i.e. γ = 0), then the groups admitting a minimal metric areprecisely the nilradicals of (standard) Einstein solvmanifolds.

Our approach proposes to vary Lie brackets rather than inner products; our tool is themoment map for the action of a reductive Lie group on the variety of all Lie algebras, whichwe show to coincide in this setting with the Ricci operator. This gives us the possibilityto use strong results from geometric invariant theory.

Riemannian foliations on H2 × R

Kyung Bai Lee University of Oklahomajoint work with S. Yi

A foliation F on a Riemannian manifold M is said to be Riemannian (=metric) if the leavesof F are locally everywhere equidistant, and homogeneous if locally, its leaves coincidewith the orbits of some group of isometries acting freely on M . As is well known, everyhomogeneous foliation is a metric foliation. The converse is known to be true on S2 ×R, Heisenberg group H2n+1. For 1-dimensional foliations, the converse is also true forconstant nonnegative-curved manifolds, while spaces of negative sectional curvature admitan abundance of non-homogeneous 1-dimensional metric foliations.

We study the product space X = H2×R with the standard product Riemannian metric.

Let F be a foliation on X. A leaf of F is said to be homogeneous if it is a (principal) orbitof a subgroup of the group of isometries. The main result is


Theorem. If a 1-dimensional metric foliation on X has a homogeneous leaf, then the

foliation itself is homogeneous.

The condition for the metric foliation having a homogeneous leaf is necessary, becausethere exist at least two kinds of 1-dimensional metric foliations on X which are not homo-geneous, (these do not contain any homogeneous leaves).

Arithmetic cusp shapes are denseDavid McReynolds University of Texas

Given a flat or almost flat manifold N , one can ask whether or not N can be smoothlyembedded or isometrically embedded as a cusp cross-section of an arithmetic real, complex,or quaternionic hyperbolic orbifold. In previous work we answered the smooth embeddingproblem. In this talk, we discuss recent work on which metrics can be realized in cusp cross-sections of arithmetic orbifolds. In particular, for flat manifolds, the flat metrics which canbe realized in the cusp cross-sections of arithmetic real hyperbolic orbifolds are dense inthe moduli space of flat metrics. Similar results hold for infra-nilmanifolds modelled onthe Heisenberg group and its quaternionic analog provided a smooth embedding exists.

Isometric actions of Heisenberg groups on compact Lorentzmanifolds

Karin Melnick University of Chicago

Connected isometry groups of compact connected Lorentz manifolds have been classifiedby Adams and Stuck and independently by Zeghib. The three non-compact, non-abeliangroups that can occur as direct factors are PSL2(R), Heisenberg groups, and certain solv-able extensions of Heisenberg groups. In the first and third cases, splitting theorems dueto Gromov and Zeghib, respectively, tell what the manifolds with these isometry groupscan be. We prove some results toward classifying compact Lorentz manifolds on whichHeisenberg groups act isometrically. The main result is a classification of actions in whichthe dimension of the Heisenberg group is one less than the dimension of the manifold.

Boundaries of systolic groupsDamian Osajda University of Wroclaw

Systolic complexes and groups (i.e. groups acting geometrically on them) were introducedby T. Januszkiewicz and J. Swiatkowski (Wroclaw) and, independently by F. Haglund(Paris) (and studied e.g. by D. Wise (USA) as combinatorial analogues of spaces of non-positive curvature. In particular they give rise to some new examples of hyperbolic (in thesense of Gromov) groups (although a class of them is different than the class of hyperbolic


groups). We will show that boundaries of hyperbolic groups constructed this way arestrongly hereditarily aspherical compacta of arbitrary dimension. This gives a new classof spaces that can occur as boundaries of hyperbolic groups (note that only few (andrelatively small) classes of spaces are known to be like that). Moreover those boundariesare interesting themselves if one studies their topology (for those that we can describeany way). We will also show that studying topology ”far away” of systolic (maybe nothyperbolic) groups allows us to distinguish them among many other groups arising ingeometric group theory. We plan to state some open problems and conjectures too.

Endomorphisms of relatively hyperbolic groupsAndrzej Szczepanski University of Gdansk

joint work with I. Belegradek

We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groupsand in particular prove the following.

• If G is a finitely generated non-elementary relatively hyperbolic group with slenderparabolic subgroups , and either G is not co-Hopfian our Out(G) is infinity, then Gsplits over a slender group.

• If a finitely generated non-parabolic subgroup H of a non-elementary relatively hy-perbolic group is not Hopfian, then H acts non-trivially on an R-tree.

• Any finitely presented group is isomorphic to a finite index normal subgroup ofOut(H) for some Kazhdan group H. (This sharpens a result of Oliver-Wise)

On a theorem of Kropholler and MislinOlympia Talelli University of Athens

P. Kropholler and G. Mislin proved that every HF group of type FP-infinity admits afinite dimensional EG-. The class HF is the smallest class of groups containing the classF of finite groups with the property : if a group G acts cellularly on a finite dimensionalcontractible CW-complex with all isotropy groups in HF, then G is in HF. EG- is theuniversal proper G-space. We show how their proof applies to a larger class of groups andhow certain algebraic invariants come out as the possible algebraic characterisation of afinite dimensional EG-.


Covolumes of Uniform Lattices acting on HyperbolicBuildings

Anne Thomas University of Chicago

Let G be a locally compact group with Haar measure µ, and Γ a uniform lattice in G, that is,a discrete cocompact subgroup. A classical question is to characterise the set of covolumesµ(Γ\G), for G an algebraic group. More recently, the question of uniform covolumes hasbeen posed for G the automorphism group of a tree. We begin the investigation of thecase where G is the automorphism group of a 2-dimensional simplicial complex, such asa hyperbolic building. In particular, when G is the automorphism group of Bourdon’sbuilding Ipq, we find the exact set of covolumes of uniform lattices in G.

Hyperbolic Coxeter groupsPavel Tumarkin Independent University of Moscow

We consider discrete groups generated by reflections in the hyperbolic space. In contrastto the spherical and Euclidean cases, hyperbolic reflection groups are not classified yet.

In this talk, we discuss a combinatorial technique giving rise to some new results con-cerning groups with relatively small number of generating reflections.

Non-injective representations of a closed surface group intoPSL(2, R)

Maxine WolffInstitut Fourier, Grenoble

Let e denote the Euler class on the space Hom(Γg, PSL(2, R)) of representations of thegroup Γg of the cosed surface Σg of genus g. Goldman showed that the connected com-ponents of Hom(Γg, PSL(2, R)) are precisely the inverse images e−1(k), for 2 − 2g ≤ k ≤2g−2, and that the components of the Euler class 2−2g and 2g−2 consist of the injectiverepresentations whose image is a discrete subgroup of PSL(2, R). We prove that non-faithful representations are dense in all the other components. We show that the image ofa discrete representation essentially determines its Euler class. Moreover, we show that forevery genus and possible corresponding Euler class, there exist discrete representations.


Poster session

On Lorentz dynamics : from group actions to warpedproducts.

Abdelouahab Arouche, Alger USTHB

We show a geometric rigidity of isometric actions of noncompact simple Lie groups onLorentz manifolds. Namely, we show that the manifold has a local warped product struc-ture of a Lorentz manifold with constant curvature by a Riemannian manifold.

On the Limit Set of Discrete Subgroups of PU(2, 1)Juan Pablo Navarrete Carrillo

It is well-known that the elements of PSL(2, C) are classified as elliptic, parabolic orloxodromic according to the dynamics and their fixed points; these three types are alsodistinguished by their trace. If we now look at the elements in PU(2, 1), then W. Goldmanintroduced the equivalent notions of elliptic, parabolic or loxodromic elements and classifiedthem by their trace. In this work we extend Goldman’s classification to all elements ofPSL(3, C); we also extend to this setting the theorem that classifies them according totheir trace. We then use this classification to study and compare two different notions ofthe limit set of a discrete subgroup of PU(2, 1). The first of these is due to Chen andGreenberg, given by thinking of these groups as automorphisms of the complex hyperbolic2-space; the second definition of the limit set is due to Kulkarni and comes from theseautomorphisms of the whole complex projective plane.

Representations of symmetry groups of carbon nanotubes andapplications

Nicolae Cotfas University of Bucharest

A single-wall carbon nanotube is a highly symmetric quasi-one-dimensional cylindricalstructure, which can be visualized as the structure obtained by rolling a honeycomb latticesuch that the endpoints of a translation vector are folded one onto the other. The symmetrygroup of nanotube depends on this vector and is one of the line groups Lqp22, L2nn/mcm.Many of the physical properties of carbon nanotube are determined by this group [1,2].

The positions of atoms forming a carbon nanotube are usually described [1,2] by usinga system of generators for the symmetry group. Each atomic position corresponds to anelement of the set Z×1, 2, . . . , n×0, 1, where n is a natural number depending on theconsidered nanotube. We obtain an alternate rather different description by starting from

Abstracts of posters 25

a description of the honeycomb lattice in terms of Miller indices. In our mathematicalmodel which is a factor space defined by an equivalence relation in the set

(v0, v1, v2) ∈ Z3 | v0 + v1 + v2 ∈ 0, 1

the neighbors of an atomic position can be described in a simpler way, and the mathematicalobjects with geometric of physical significance have a simpler and more symmetric form[3-6].

References.

[1] M. Damnjanovic et al., Symmetry and lattices of single-wall nanotubes, J. Phys. A: Math. Gen., 32(1999), 4097-4104.

[2] M. Damnjanovic et al., Wigner-Eckart theorem in the inductive spaces and applications to optical transitions in nano-

tubes, J. Phys A: Math. Gen., 37(2004), 4059-4068.

[3] N. Cotfas, an alternate mathematical model for single-wall carbon nanotubes, J. Geom. Phys., accepted (math-ph/0403011).

[4] N. Cotfas, Quantum random walks on carbon nanotubes and quasicrystals, J. Phys. A: Math. Gen., 34(2001), 5469-5483.

[5] N. Cotfas, Random walks on carbon nanotubes and quasicrystals, J. Phys. A: Math. Gen., 33(2000), 2917-2927.

[6] http://fpcm5.fizica.unibuc.ro/~ncotfas

Translations in simply transitive groups of affine motionsTine De Cat K.U.Leuven Campus Kortrijk

joint work with K. Dekimpe

We study simply transitive affine actions of nilpotent Lie groups G on Rn. Motivated by

a conjecture formulated by Auslander, we investigate, in case G admits such an action,whether or not there necessarily exists a nontrivial subgroup of G acting via pure transla-tions. We show that in case the Lie group G has a 1-dimensional commutator subgroup,then for any simply transitive affine action of G, there is indeed a nontrivial subgroup of Gacting by pure translations. This result no longer holds in case the commutator subgroupis higher dimensional. We also determine all five-dimensional nilpotent Lie groups actingsimply transitively and affinely on R

5 in such a way that only the identity element acts asa pure translation. In fact, a complete classification of all possible such actions is obtained.Finally we also investigate whether the conjecture of Auslander holds in case the Lie groupis free 2-step nilpotent and in case the Lie group is filiform.

The Anosov relation for Nielsen numbers of maps oninfra-nilmanifolds

Bram De Rock K.U.Leuven Campus Kortrijkjoint work with K. Dekimpe and W. Malfait

In fixed point theory, the Nielsen number N(f) and the Lefschetz number L(f) are twonumbers associated with a continuous self-map f : M → M of a smooth closed manifoldM to provide information on the fixed points of f . N(f) gives more information, butunfortunately N(f) is not readily computable from its definition while L(f) is much easier


to calculate.A celebrated theorem of Anosov states that for any continuous map f : M → M of anilmanifold M , one has that N(f) = |L(f)|. He also showed that this result does nothold in general for infra-nilmanifolds, since he constructed a counterexample on the Kleinbottle.We show that Anosov’s theorem is still true for infra-nilmanifolds with odd order holonomygroup. We also establish a necessary and sufficient condition for expanding maps f ofinfra-nilmanifolds M in order that the Anosov relation should hold for this specific maps.Namely, N(f) = |L(f)| if and only if M is orientable.

The solution of a length five equation over groupsAnastasia Evangelidou Larnaka, Cyprus

Let G be a group, t an unknown and r(t) an element of the free product G ∗ 〈t〉. Theequation r(t) = 1 has a solution over G if it has a solution in a group H containing G.The Kervaire-Laudenbach (KL) conjecture asserts that if the exponent sum of t in r(t) isnon-zero the equation has a solution.

There have been several results concerning the KL conjecture when the problem isrestricted to a type of group (e.g. the group is locally indicable or locally residually finite[5], [6] and [9]). Also the conjecture has been studied for certain types of equations, forexample for equations of certain length. In particular it has been proved that the KLconjecture is true for equations up to length five ([1], [2], [3], [4], [7] and [8]).

An equation of length five can be put into one of the following forms by cyclic per-mutation and inversion: r0(t) = atbtctdtet = 1, r1(t) = atbtctdtet−1 = 1 , r2(t) =atbtctdt−1et−1 = 1, r3(t) = atbtct−1dtet−1 = 1.

The aim of this presentation is to illustrate the methodology used for the solution ofthe equation r(t) = 1 that uses curvature arguments on relative diagram and the weighttest on the start graph Γ of the equation.

References.

[1] Evangelidou, A. Equations of length five, Ph. D. Thesis, University of Nottingham 2003.

[2] Edjvet, M. A Singular equation of length four over groups, Algebra Colloquium 2000, 7(3), 247-274.

[3] Edjvet, M.; Howie, J. The solution of length four equations over groups, Transactions of the American Mathematical

Society 1991, 326, 345-369.

[4] Edjvet, M.; Juhasz, A. Equations of length 4 and one-relator products, Math. Proc. Camb. Phil. 2000, 129, 217-229.

[5] Gersten, S. M. Reducible diagrams and equations over groups, In Essays in Group Theory (8), Gersten, S. M. Ed.; MSRI

publications: Springer-Verlag, 1987, 15-73.

[6] Howie, J. On pairs of 2-complexes and systems of equations over groups, J. Reine Angew. Math. 1981, 324, 165-174.

[7] Howie, J. The solution of length three equations over groups, Proceedings of the Edinburgh Mathematical Society 1983,

26, 89-96.

[8] Levin. F., Solutions of equations over groups, Bull. Amer. Math. Soc. 1962, 68, 603-604.

[9] Rothaus, S. O. On the non-triviality of some group extensions given by generators and relators, Annals of Mathematics

1977, 106(2), 599-612.

Abstracts of posters 27

Coxeter decompositions of polytopesAnna Felikson Independent University of Moscow

Let P be a polytope in Sn, En or Hn. P is called a Coxeter polytope if all dihedral anglesof P are submultiples of π. A Coxeter decomposition of P is a tiling of P by finite numberof Coxeter polytopes such that if two tiles of the decomposition have a common facet thenthese tiles are symmetric to each other with respect to this facet.

Coxeter decomposition of a Coxeter polytope corresponds to a reflection subgroup ofthe reflection group generated by the reflections with respect to the facets of the poly-tope. Decompositions of non-Coxeter polytopes correspond to non-standard systems ofgenerators for the same group.

We discuss general properties of Coxeter decompositions and decompositions of thepolytopes with low numbers of facets.

Groupes moyennables, dimension topologique moyenne etsous-decalages

Fabrice Krieger IRMA, Strasbourg

Un theoreme du a Jaworski affirme que toute action continue minimale d’un groupe com-mutatif G sur un espace compact metrisable X de dimension topologique finie, se plongedans le G-decalage sur [0, 1]G. Pour G = Z, E. Lindenstrauss et B. Weiss ont montre qu’onne pouvait pas supprimer l’hypothese de finitude de la dimension topologique de X enconstruisant un contre-exemple. Nous allons generaliser le resultat de Lindenstrauss-Weisset demontrer que si G est un groupe infini denombrable, moyennable et residuellement fini,alors il existe un espace compact metrisable X, muni d’une action continue minimale deG, tel que le systeme dynamique obtenu ne se plonge pas dans le G-decalage sur [0, 1]G.Cela montre en particulier que l’on ne peut pas supprimer l’hypothese de finitude de la di-mension topologique de X dans le theoreme de Jaworski lorsque G est de type fini. L’outilessentiel utilise dans la demonstration du resultat est la dimension topologique moyennequi est un invariant topologique des actions de groupes moyennables introduit par M. Gro-mov. En utilisant cet invariant, on reduit le probleme a la construction d’un G-systememinimal X de dimension topologique moyenne plus grande que celle du G-decalage [0, 1]G.

Commutative polarizationsAlfons Ooms Limburgs Universitair Centrum

Let L be a finite dimensional Lie algebra over a field k of characteristic zero and let U(L)be its enveloping algebra with quotient division ring D(L). Let P be a Lie subalgebra ofL. A necessary and sufficient condition is given in order for D(P ) to be a maximal subfield


of D(L). This settles a question by Jacques Alev. This condition is satisfied if P is acommutative polarization (CP) of L and the converse holds in case L is algebraic.

The purpose of this talk is to study Lie algebras admitting a CP and to demonstratetheir widespread occurrence. In particular, we will look at the corresponding inducedrepresentations and their kernels, the primitive ideals of U(L). Special attention is devotedto the situation where L is a semi direct product S ⊕ P , where P is a commutative idealof L. For instance, let k be algebraically closed and let S be a simple Lie algebra, actingirreducibly on P . Then the above condition is satisfied if and only if dimS < dimP .Finally, let N be the nilradical of a parabolic Lie subalgebra of a simple Lie algebra Lof type An or Cn and suppose k is algebraically closed. Then, in cooperation with A.Elashvili, it is proved that N has a commutative polarization. As a bonus we obtain anexplicit formula for the index i(N) of N .

List of participants 29

List of participants

Herbert Abels Fakultat fur Mathematik, Universitat BielefeldPostfach 100131, D-33501 Bielefeld, [email protected]

Kristina Altmann Technische Universitat DarmstadtFachbereich Mathematik, AG5, Schlossgassenstr. 7, 64289 Darmstadt, [email protected]

Sergei Antonyan Universidad Nacional Autonoma de MexicoDept. Mat., Facultad de Ciencias, UNAM, 04510 Mexico D.F., [email protected]

Abdelouahab Arouche USTHB AlgerBP 32 USTHB 16123 Bab-Ezzouar Alger, [email protected]

Thierry Barbot CNRS / ENS LyonENS Lyon UMPA, 46 Allee d’Italie, 69364 Lyon, [email protected]

Oliver Baues Universitat KarlsruheMathematisches Institut II, Englerstr. 2, 76128 Karlsruhe, [email protected]

Yves Benoist ENS Paris45 Rue d’Ulm, 75005 Paris, [email protected]

Adalbert Bovdi University of DebrecenInstitute of Mathematics and Informatics, University of Debrecen, P.O.Box 12,Debrecen H-4010, [email protected]

Victor Bovdi University of DebrecenInstitute of Mathematics and Informatics, University of Debrecen, P.O.Box 12,Debrecen H-4010, [email protected]

Emmanuel Breuillard IHES35 Route de Chartres, 91440 Bures-sur-Yvette, [email protected]


Martin Bridson Imperial College LondonMathematics, Huxley Building, Imperial College London, London SW7 2AZ,England, [email protected]

Dietrich Burde Universitat WienFak. Mathematik, Nordbergstr. 15, 1090 Wien, [email protected]

Pierre-Emmanuel Caprace Universite Libre de BruxellesULB CP216, Boulevard du Triomphe, 1050 Bruxelles, [email protected]

Virginie Charette University of ManitobaDept. of Mathematics, University of Manitoba, Winnipeg MB R3T2N2, [email protected]

Suhyong Choi KAISTDaejeon, Yuseong-Gu Guseong 305-701, South [email protected]

Nicolae Cotfas University of BucharestP.O. Box 76-54, Post Office 76, Bucharest, [email protected]

Pallavi Dani University of Chicago5135 S. University Avenue, #3B, Chicago, IL 60615, United States of [email protected]

Tine De Cat K.U.Leuven campus KortrijkEtienne Sabbelaan 53, B-8500 Kortrijk, [email protected]

Yves de Cornulier EPF LausanneEPFL, Dept. de Mathematiques, Dat. MA, CH1015 Lausanne, [email protected]

Karel Dekimpe K.U.Leuven campus KortrijkEtienne Sabbelaan 53, B-8500 Kortrijk, [email protected]

Pierre de la Harpe Universite de GeneveSection de Mathematiques, CP64, 1211 Geneve 4, [email protected]


Bram De Rock K.U.Leuven campus KortrijkEtienne Sabbelaan 53, B-8500 Kortrijk, [email protected]

Sandra Deschamps K.U.Leuven campus KortrijkEtienne Sabbelaan 53, B-8500 Kortrijk, [email protected]

Franki Dillen K.U.LeuvenCelestijnenlaan 200B, B-3001 Heverlee, [email protected]

Anastasia EvangelidouEvripidou 39, Larnaka 6036, [email protected]

Benson Farb University of Chicago5734 S. University Avenue, Chicago IL 60637, United States of [email protected]

Yves Felix Universite Catholique de LouvainBatiment Marc de Hemptinne, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve,[email protected]

Anna Felikson MPIM Bonn / Independent University of MoscowMPI for Mathematics, Vivatsgasse 7, D-53111 Bonn, [email protected]

Oscar Garcia-Prada CSIC MadridInstituto de Matematicas y Fısica Fundamental, Consejo Superior de InvestigacionesCientıficas, Serrano, 113 bis, 28006 - Madrid, [email protected]

Etienne Ghys ENS LyonENS Lyon UMPA, 46 Allee d’Italie, 69364 Lyon, [email protected]

William Goldman University of MarylandUniversity of Maryland, College Park, MD 20742, United States of [email protected]

Fritz Grunewald Universitat DuesseldorfMathematisches Institut, Heinrich Heine Universitat Dusseldorf, [email protected]


Mohammed Guediri King Saud UniversityDept. of Mathematics, College of Science, King Saud University, P.O.Box 2455,Riyadh 11451, Saudi [email protected]

Antonin Guilloux Ecole Normale Superieure45 Rue d’Ulm, 75005 Paris, [email protected]

Toshiaki Hattori Tokyo Institute of TechnologyDept. of Mathematics, Oh-okayama, Meguro, Tokyo 152-8551, [email protected]

Paul Igodt K.U.Leuven campus KortrijkEtienne Sabbelaan 53, B-8500 Kortrijk, [email protected]

Delarem Kahrobaei University of St. AndrewsMathematical Institute (Room 315), University of St. Andrews, North Haugh,St. Andrews, Fife, KY 16 9SS, Scotland, [email protected]

Yoshinobu Kamishima Tokyo Metropolitan UniversityDept. of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Ha-chioji, Tokyo 192-0397 , [email protected]

Caroline Keil Universitat DusseldorfMathematisches Institut, Heinrich Heine Universitat Dusseldorf, [email protected]

Inkang Kim Seoul National UniversitySeoul National University, Math. Dept., 151-342 Seoul, South [email protected]

Ji-Ae Kim Seoul National UniversitySeoul National University, Math. Dept., 151-342 Seoul, South [email protected]

Benjamin Klopsch Universitat DuesseldorfMathematisches Institut, Heinrich Heine Universitat Dusseldorf, [email protected]

Fabrice Krieger IRMA StrasbourgIRMA, UMR 7501 CNRS/ULP, 7 rue Rene Descartes, 67084 Strasbourg Cedex,[email protected]


Jorge Lauret FAMAF, Universidad Nacional de CordobaHaya de la Torre s/n, 5000 Cordoba, [email protected]

Kyung Bai Lee University of OklahomaDepartment of Mathematics, University of Oklahoma, Norman, OK 73019, UnitedStates of Americakb [email protected]

Alexander Lichtman University of Wisconsin-ParksideKenosha, WIS314, United States of [email protected]

Seonhee Lim ENS Paris / Yale University45 Rue d’Ulm, 75005 Paris, [email protected]

Gabriele Link Universitat KarlsruheMathematisches Institut II, Englerstr. 2, 76128 Karlsruhe, [email protected]

Isabelle Liousse Universite de Lille 1Laboratoire Painleve, Universite de Lille 1, 59655 Villeneuve d’ascq cedex, [email protected]

Wim Malfait K.U.Leuven campus KortrijkEtienne Sabbelaan 53, B-8500 Kortrijk, [email protected]

Alec Mason Glasgow UniversityDept. of Mathematics, Glasgow G12 P9W, Scotland, [email protected]

Ben McReynolds University of Texas4208 Avenue B, Austin TX 78751, United States of [email protected]

Karin Melnick University of Chicago5734 S. University Avenue, Chicago IL 60637, United States of [email protected]

Nicolas Michelacakis University of CyprusMathematics and Statistics Department, University of Cyprus, P.O. Box 20537,Nicosia 1678, [email protected]


Igor Mineyev University of Illinois at Urbana-ChampaignDept. of Mathematics, UIUC, Urbana IL61801, United States of [email protected]

Bernhard Muhlherr Universite Libre de BruxellesULB CP216, Boulevard du Triomphe, 1050 Bruxelles, [email protected]

Juan Pablo Navarrete Carrillo Universidad Nacional Autonoma de MexicoDept. Mat., Facultad de Ciencias, UNAM, 04510 Mexico D.F., [email protected]

Guennadi Noskov Fakultat fur Mathematik, Universitat BielefeldPostfach 100131, D-33501 Bielefeld, [email protected]

Alfons Ooms Limburgs Universitair CentrumDept. WNI E87, Limburgs Universitair Centrum, Universitaire Campus, B-3590Diepenbeek, [email protected]

Damian Osajda University of WroclawKielczowska 51b/2, 51-315 Wroclaw, [email protected]

John Panagopoulos University of AthensUniversity of Athens, Panepistimiopolis, Athens 15784, [email protected]

Krzystof Pawalowski Adam Mickiewick UniversityUl. Umultowska 87, 61614 Poznan, [email protected]

Hannes Pouseele K.U.Leuven campus KortrijkEtienne Sabbelaan 53, B-8500 Kortrijk, [email protected]

Bartosz Putrycz University of GdanskInstytut Matematyki UG, ul. Wita Stwosza 57, 80-952 Gdansk, [email protected]

Andrzei Szczepanski University of GdanskInstytut Matematyki UG, ul. Wita Stwosza 57, 80-952 Gdansk, [email protected]


Olympia Talelli University of AthensUniversity of Athens, Panepistimiopolis, Athens 15784, [email protected]

Anne Thomas University of Chicago5734 S. University Avenue, Chicago IL 60637, United States of [email protected]

Paulo Tirao FAMAF, Universidad Nacional de CordobaHaya de la Torre s/n, 5000 Cordoba, [email protected]

Domingo Toledo IHES35 Route de Chartres, 91440 Bures-sur-Yvette, [email protected]

Pavel Tumarkin MPIM BonnMPI for Mathematics, Vivatsgasse 7, D-53111 Bonn, [email protected]

Joost Van Hamel K.U.LeuvenCelestijnenlaan 200B, B-3001 Heverlee, [email protected]

Stephen Wang University of Chicago5510 S. Wood Lawn #402, United States of [email protected]

Cynthia Will FAMAF, Universidad Nacional de CordobaHaya de la Torre s/n, 5000 Cordoba, [email protected]

Maxine Wolff Institut Fourier, Grenoble100, rue des Maths, BP74 38402 St Martin d’Heres, [email protected]


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, Fundamental methods andTechniques in Mathematics

• the Universite Catholique de Louvain (UCL), and

• the Katholieke Universiteit Leuven (Campus Kortrijk)

for their support.

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