Discrete groups and Geometric Structures with Applications

Conference Booklet of the 4th Kortrijk Workshop

Discrete groups and Geometric Structures with Applications Oostende, May 31 - June 3, 2005

Karel Dekimpe, Paul Igodt and Hannes Pouseele organizers

Yves F´elix, William Goldman, Fritz Grunewald, Paul Igodt and Kyung Bai Lee scientific committee

Conference Booklet

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Dear participant, we wish you a very warm welcome on this fourth edition of the Kortijk workshop on Discrete 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 mathematical and a personal level. Karel Dekimpe, Paul Igodt and Hannes Pouseele

Schedule

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Schedule Invited 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 parallel sessions and the poster session, there are also 4 plenary sessions of 45 minutes each. All plenary sessions take place in the auditorium. The parallel sessions take place in the rooms Iridia 1 (IR1) and Iridia 3 (IR3).

Tuesday, May 31 Morning session 1 08.30-09.30 09.30-09.45 09.45-10.45 10.45-11.15 11.15-12.15

Registration Welcome and Opening of the Workshop by P. Igodt. E. Ghys: Quasi-morphisms on SL(2, Z): old and new Coffee break O. Baues: Homotopy equivalences of solv-manifolds and arithmeticity

Afternoon session 1 14.00-14.25 IR1 D. Osajda: Boundaries of systolic groups IR3 K. Dekimpe: The Auslander conjecture for NIL-affine crystallographic groups 14.35-15.00 IR1 T. Barbot: On the dynamics of surface groups on the flag variety IR3 I. Kim: On Marcus conjecture 15.10-15.35 IR1 E. Breuillard: Uniform versions of the Tits alternative IR3 M. Guediri: A new class of compact spacetimes without closed nonspacelike geodesics 15.40-16.10 Coffee break 16.10-16.35 IR1 H. Abels: A question of C.L. Siegel IR3 A. Thomas: Covolumes of Uniform Lattices acting on Hyperbolic Buildings 16.45-17.30 G. Link: Geometry of discrete subgroups of higher rank Lie groups

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Wednesday, June 1 Morning session 2 09.00-10.00 D. Toledo: Non-arithmetic uniformizations of some real moduli spaces 10.00-10.30 Coffee break 10.30-11.30 Y. Benoist: 3-dimensional projective tilings 11.40-12.05 IR1 K. Altmann: Hyperbolic lines in unitary space IR3 S. Choi: Projective structures on 3-dimensional orbifolds and some deformations Afternoon session 2 14.00-14.25 IR1 J. Lauret: A canonical compatible metric for geometric structures on nilmanifolds IR3 Y. de Cornulier: Finitely presented wreath products 14.35-15.00 IR1 D. Burde: Classical Yang-Baxter Equation and Geometry of Lie Groups IR3 N. Cotfas: Permutation representations defined by G-clusters with application to quasicrystal physics 15.10-16.00 Poster Session 16.00-16.30 Coffee break 16.30-17.15 H. Pouseele: Betti number behavior for nilpotent Lie algebras

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Thursday, June 2 Morning session 3 09.00-10.00 B. Farb: Problems and progress in understanding the Torelli group 10.00-10.30 Coffee break 10.30-10.55 IR1 M. Wolff: Non-injective representations of a closed surface group into P SL(2, R) IR3 Y. Kamishima: Cusp cross-sections of hyperbolic orbifolds by Heisenberg nilmanifolds 11.05-11.30 IR1 P. Dani: Finding the density of finite order elements in infinite groups IR3 B. McReynolds: Arithmetic cusp shapes are dense 11.40-12.25 I. Mineyev: Cohomology and hyperbolicity Afternoon Social activity / Excursion Evening Conference Dinner

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Friday, June 3 Morning session 4 09.00-10.00 O. Garcia–Prada: G-Higgs bundles and surface group representations 10.00-10.30 Coffee break 10.30-11.30 M. Bridson: Limit groups: non-positive curvature, group theory and logic 11.40-12.05 IR1 V. Charette: Affine deformations of the holonomy group of a three-holed sphere IR3 A. Szczepanski: Endomorphisms of relatively hyperbolic groups Afternoon session 3 14.00-14.25 IR1 K. Melnick: Isometric actions of Heisenberg groups on compact Lorentz manifolds IR3 O. Talelli: On a theorem of Kropholler and Mislin 14.35-15.00 IR1 K.B. Lee: Riemannian foliations on H2 × R IR3 P. Tumarkin: Hyperbolic Coxeter groups 15.10-15.40 Coffee break 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

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Invited talks Homotopy equivalences of solv-manifolds and arithmeticity Oliver Baues Universit¨at Karlsruhe To the fundamental group of every solv-manifold M and, more generally, to every torsionfree polycyclic by finite group we associate a linear algebraic group which is defined over the rational numbers. This construction is functorial and determines many geometric properties of the K(π, 1)-space M . In this talk, we explain that the group of homotopy classes of self-equivalences of M is isomorphic to an arithmetic group. 3-dimensional projective tilings Yves Benoist ENS, Paris We will study the following setting: Ω is a convex open subset of the real projective 3-space with a cocompact discrete group of projective transformations Γ. For instance, we will see that, when Ω is indecomposable and Γ torsion free, the properly embedded flat triangles in Ω project in the quotient M := Γ\Ω onto finitely many disjoint tori and Klein bottles which induce an atoroidal decomposition of M . Limit groups: non-positive curvature, group theory and logic Martin Bridson Imperial College, London In his solution to the Tarski problem, Zlil Sela described the class of “limit groups”. Limit groups have Cayley graphs that arise as pointed Gromov-Hausdorff limits of Cayley graphs of free groups. Such groups L are precisely those with the property that, given any finite subset S ⊂ L, there is a homomorphism to a free group that is injective on S. They can also be characterised by their first order logic, and in terms of the structure of their classifying spaces. In this talk I shall outline the basic structure of limit groups (following Sela), describe the geometry of their classifying spaces, and then present recent work by Howie and I on the subgroup structure of limit groups and, more strikingly, their direct products.

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Discrete Groups and Geometric Structures II Problems and progress in understanding the Torelli group Benson Farb University of Chicago

The Torelli group T (S) is defined to be the subgroup of the mapping class group consisting of those mapping classes acting trivially on H1 (S, Z). The study of T (S) connects to 3-manifold theory, symplectic representation theory, combinatorial group theory, and algebraic geometry. In this talk I will describe some of the main themes in this beautiful topic, concentrating on the combinatorial and (co)homological aspects. I will also describe some recent progress in this direction, as well as a number of open questions and conjectures. Different parts of the talk will describe joint work with (separately) Daniel Biss, Nick Ivanov and Tara Brendle. G-Higgs bundles and surface group representations Oscar Garcia-Prada CSIC, Madrid In this talk we describe the theory of G-Higgs bundles over a compact Riemann surface where G is a non-compact real reductive Lie group. We then show how this theory is used to study the moduli space of representations of the fundamental group of the surface in G. 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 new Etienne Ghys ENS, Lyon A quasi-morphism on a group G is a map f from G to R which is a “morphism up to a bounded error”, ie such that the modulus of f (xy) − f (x) − f (y) is uniformly bounded. In the first part of my talk, I plan to explain why these objects are worth studying in relation with dynamics and topology. Then I will consider the case where G is the modular group SL(2, Z) and give many examples. Some of these examples are very old but some others are pretty recent. Non-arithmetic uniformizations of some real moduli spaces Domingo Toledo IHES, Paris This lecture will describe joint work with Allcock and Carlson on the structure of the real points 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, but not necessarily in an obvious way. Namely, there may not be a single anti-holomorphic involution of complex hyperbolic space whose fixed point set uniformizes the real moduli

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space. This will be illustrated with the example of polynomials of degree 6, where the complex moduli space is known to be uniformized by complex hyperbolic 3-space. The moduli space of real polynomials turns out to be real hyperbolic, containing 4 pieces that parametrize the spaces of real polynomials with each of the 4 different possible configurations of real and complex conjugate pairs of roots. While each of the pieces is uniformized by an arithmetic group, the group of the whole space is non-arithmetic, in the spirit of a construction of Gromov and Piatetski-Shapiro. A more involved example of the same phenomenon is the moduli space of real cubic surfaces.

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Plenary talks Proper affine actions and geodesic flows of hyperbolic surfaces William Goldman University of Maryland joint work with F. Labourie and G. Margulis Let F be a Schottky group inside SO(2, 1), and let S = H 2 /F be the corresponding hyperbolic surface. Let GC denote the space of geodesic currents on S. The cohomology group H 1 (F, V ) parametrizes equivalence classes of affine deformations of F acting on an irreducible representation V of SO(2, 1). We define a continuous biaffine map M : GC × H 1 (F, V ) → R which is linear on the vector space H 1 (F, V ). An affine deformation F[u] corresponding to a cohomology class [u] in H 1 (F, V ) acts properly if and only if M (c, [u]) is nonzero for all geodesic currents c. Consequently the set of proper affine actions whose linear part is a Schottky group identifies with a bundle of open convex cones in H 1 (F, V ) over the Teichmueller space of S. Geometry of discrete subgroups of higher rank Lie groups Gabriele Link Universit¨at Karlsruhe Let X = G/K be a higher rank symmetric space of noncompact type, ∂X its geometric boundary, and Γ ⊂ G a discrete group. In this talk we are going to investigate the structure of the geometric limit LΓ := Γx ∩ ∂X in purely geometrical terms. We will describe the dynamics of axial isometries which, together with an approximation argument originally due 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 discrete groups (see i.e. [1], [2]), the advantage of our proof is its purely geometric nature which allows 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 the geometric boundary. As an application, we describe a new construction of free groups in higher rank symmetric spaces.

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References. [1] Y. Benoist, Propri´ et´ es asymptotiques des groupes lin´ eaires 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, Sankhy˜ a 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 hyperbolicity Igor Mineyev University of Illinois I will give a review of cohomological properties of hyperbolic groups, in particular the Gersten’s characterization of hyperbolic groups by the vanishing of the second `∞ cohomology with `∞ coefficients. We will discuss recent developments in this area and new methods used: a characterization of hyperbolicity by `∞ cohomology with various other coefficients and cohomological properties of relatively hyperbolic groups. Betti number behaviour for nilpotent Lie algebras Hannes Pouseele K.U.Leuven Campus Kortrijk In general, it is quite a feat to determine the Betti numbers of all Lie algebras of a certain type. 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. In contrast to this combinatorial approach, I use arguments of spectral sequence type to determine an explicit formula for the Betti numbers of low-dimensional split extensions of a Heisenberg Lie algebra. Amongst others, the family of the twisted standard filiform Lie algebras is contained in this class. Numerous examples, including all algebras with an abelian ideal of codimension 1, show a unimodal behavior, that is, βp < βp+1 for all p less or equal to half the dimension of the algebra. This behavior is by no means a general phenomenon. We (joint work with Dietrich Burde) use the explicit formulas developed above to show that the class of one-dimensional extensions of a Heisenberg Lie algebra already contains examples, indeed whole families of examples, with a non-unimodal Betti number distribution. The same kind of spectral sequence argument turns out to be useful in the context of the Toral Rank Conjecture, stating that the total dimension of the cohomology of a nilpotent Lie algebra is bounded below by 2z , where z is the dimension of the center. For split metabelian Lie algebras we (joint work with Paulo Tirao) construct an explicit embedding of the cohomology of the center into the cohomology of the algebra, thus proving the Toral Rank Conjecture for this class of algebras. This class of split metabelian algebras already contains a family that doesn’t fit into the classical – combinatorial – approach of the Toral Rank Conjecture.

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Short Communications A question of C.L. Siegel Herbert Abels Universit¨at Bielefeld Let Γ = SL(n, Z) and G = SL(n, R). The main result of reduction theory describes a fundamental set S for Γ in G, i.e., a subset S of G such that the natural map G → Γ\G restricted to S, let us call it π : S → Γ\G, is surjective and has finite fibers. The set S described in reduction theory is called a Siegel set. Siegel asked in 1959 if Γ has the additional property of preserving the natural metric up to a constant. In recent work with G.A. Margulis we gave a positive answer to this question. Our solution has three interesting features: 1. It works for all reductive groups G over local fields and their corresponding arithmetic subgroups. Positive answers to Siegels question had been known before for reductive groups 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 symmetric space (as considered by the authors above) or from the Bruhat-Tits building, but also for word metrics. 3. The proof gives additional information in reduction theory.

Hyperbolic lines in unitary space Kristina Altmann Technische Universit¨at 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 characterisations. The game is the following. Choose a graph ∆, and try to identify all connected graphs which are locally ∆. Let F be a finite field, F 6= F2 , and K be a quadratic extension. We focus on the n-dimensional vector space Vn over K endowed with a non-degenerate hermitian form and define the graph S(Vn ) on the hyperbolic lines of Vn where two hyperbolic lines l and m are 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 to S(Vn+2 ). To obtain that result we first study graphs isomorphic to S(Vn ) in order to recover the geometry of Vn from the graph S(Vn ). Next we prove that for n ≥ 8 the diameter of the graph Γ is 2 and for n = 7 the diameter of the graph Γ is 3. This global property together

Abstracts of short communications

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with the local properties we have studied before enables us to determine the isomorphism type 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 the graph S(V8 ) also the graph on the fundamental SL2 ’s of the group 2 E6 (K) with commuting as 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 variety Thierry Barbot ENS, Lyon Let Γ be the fundamental group of a closed surface with genus g > 1. Any morphism of Γ in SL(2, R) induces a morphism ρ0 : Γ → G, where G ≈ SL(3, R) is the group of projective transformations of the projective plane. The induced projective action of Γ preserves a point x0 and a projective line d0 . The flag variety X is the set of pairs (x, d) (a flag) where x is a point of the projective plane, and d a projective line containing x. The differential of any 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) where x belongs to d0 and d contains x0 . A fundamental observation is that ρ0 is a (G, Y )-Anosov representation, in the meaning of F. Labourie, where Y is the space of triples of distinct points in RP 2 . It follows that for morphisms ρ : Γ → G near to ρ0 , the induced action on X still preserves a topological circle L(ρ) in X, which, in general, is only H¨older continuous. More precisely, if L(ρ) is Lipschitz continuous, then its is analytic, and ρ(Γ) preserves a point or a projective line in 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 one important feature: there are two invariant annuli, and the dynamic on these invariant annuli highly depends on the morphism ρ. As a corollary, we can show that the action of ρ(Γ) on X is topologically conjugate to the action of ρ(Γ0 ) if and only if the representations ρ and ρ0 are conjugate inside G. Uniform versions of the Tits alternative Emmanuel Breuillard IHES, Paris The celebrated Tits alternative states that a finitely generated non-virtually solvable linear group Γ contains a non-abelian free subgroup. We show that, for any finite generating set S of Γ, generators of such a free subgroup can be found as words of length at most r in

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S, where r = r(Γ) is a constant depending on Γ only. This implies uniform exponential growth as well as uniform non-amenability for finitely generated non virtually solvable linear groups in arbitrary characteristic and thus subsumes the recent work of EskinMozes-Oh. Although the constant r cannot be made entirely independent of Γ, we show that 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 entropy of discrete subgroups. Analogous results are obtained for the class of finitely generated solvable groups.

Classical Yang-Baxter Equation and Geometry of Lie Groups Dietrich Burde Universit¨at Wien Let G be a Lie group with Lie algebra g. Any skewsymmetric element r ∈ Λ2 (g) satisfying the CYBE c(r) = [r 12 , 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-one correspondance between the r-matrices in G and the set of connected Lie subgroups of T ∗ G which carry a left-invariant affine structure and whose Lie algebras are Lagrangian graphs in g ⊕ g∗ . This is proved in [1]. We study a natural generalization of the notion of classical r-matrices by replacing the adjoint action involved by an arbitrary g-module action. Then one obtains left-invariant affine structures on Lie groups in much greater generality. Our results 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-holed sphere Virginie Charette University of Manitoba joint work with T.A. Drumm and W.M. Goldman Suppose Γ = hγ1 , γ2 i 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 the associated closed geodesic. This function can be extended to parabolic transformations. If two transformations are of opposite sign then no group containing both of them can act properly on E.

Abstracts of short communications

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Suppose γ1 and γ2 are both hyperbolic and consider the invariant lines of their linear parts in the hyperbolic plane. When these lines are ultraparallel, Jones showed that if the sign of the generators and the sign of their product are the same then the group acts properly discontinuously on E. When the lines intersect, checking the sign of any finite number 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 space of Γ. Projective structures on 3-dimensional orbifolds and some deformations Suhyoung Choi KAIST We discuss some examples of 3-dimensional polyhedral orbifolds with projective structures and their deformations. The techniques are from the higher dimensional Kac-Vinberg construction taken to the lower dimensions by taking 3-dimensional sections of the ndimensional simplices.

Permutation representations defined by G-clusters with application to quasicrystal physics Nicolae Cotfas University of Bucharest Quasicrystals are systems with long-range order but non-periodic. Atomic structure determination 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 invariant under a finite group G, and the high-resolution electron microscopy suggests that the quasicrystal can be regarded as a quasiperiodic packing of interpenetrated copies of a well-defined G-invariant atomic cluster. From a mathematical point of view, the G-cluster C describing the microstructure of a quasicrystal is a finite union of orbits of G. By starting from the action of G on C we can define a permutation representation of G in a higher-dimensional Euclidean space Ek = (Rk , h·, ·i). 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 + E obtained by translating the hypercube [0, 1]k along E. The set Q is a quasiperiodic packing of interpenetrating copies of C and can be regarded as a mathematical model for the considered quasicrystal [3-8].

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References. [1] [2] [3] [4] [5] [6] [7] [8]

A. Katz and M. Duneau, J. Phys. (France), 47(1986), 181. V. Elser, Acta Cryst. A, 42 (1986), 36. N. Cotfas and J.-L. Verger-Gaugry, J. Phys. A: Math. Gen., 30(1997), 4283. N. Cotfas, Lett. math. Phys., 47(1999), 111. N. Cotfas, J. Phys. A: Math. Gen., 32(1999), 8079. N. Cotfas, J. Phys. A: Math. Gen., 37(2004), 3125. N. Cotfas, Ferroelectrics, 305(2004), 33. http://fpcm5.fizica.unibuc.ro/~ncotfas

Finding the density of finite order elements in infinite groups Pallavi Dani University of Chicago Consider a finitely generated infinite group Γ. Let P be a property that elements of Γ might have, such as having finite order, being contained in a maximal abelian subgroup or being self-centralizing. What is the probability that a given element of Γ has the property P ? Fix a finite generating set S for Γ and let B(r) and E(r) denote the ball of radius r and the set of elements with property P in the ball of radius r respectively. Define |E(r)| , if it exists. FP (Γ, S) measures the density of elements in Γ with FP (Γ, S) = lim r→∞ |B(r)| the property P . We consider the case when P is the property of having finite order and compute this limit for a number of examples, including virtually nilpotent groups and word hyperbolic groups, in which very different phenomena occur. Finitely presented wreath products Yves de Cornulier EPF Lausanne Proposition Let N 6= 1 and G be finitely presented groups, and X a G-set. Then the wreath 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 X 2 (for the diagonal action). This leads to the question: for what finitely generated groups G does there exist an infinite G-set X satisfying (i)-(ii). We present some examples and some obstructions.

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The Auslander conjecture for NIL-affine crystallographic groups Karel Dekimpe K.U.Leuven Campus Kortrijk Joint 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 ) = N o Aut(N ) acting properly discontinuously and cocompactly on N . This situation is a natural generalization of the so-called affine crystallographic groups. We prove that for all dimensions 1 ≤ n ≤ 5 the generalized Auslander conjecture holds, i.e., that such subgroups are virtually polycyclic. A new class of compact spacetimes without closed nonspacelike geodesics Mohammed Guediri KSU, Riyadh Using simply transitive affine actions of the three-dimensional Heisenberg group H 3 on R3 , we construct the first examples of geodesically complete compact spacetimes with regular globally hyperbolic coverings but without closed nonspacelike geodesics. Cusp cross-sections of hyperbolic orbifolds by Heisenberg nilmanifolds Yoshinobu Kamishima Tokyo Metropolitan University We study the geometric boundary problem. Long and Reid have shown that some compact flat 3-manifold cannot be diffeomorphic to a cusp cross-section of a 1-cusped finite volume hyperbolic manifold. Similar to the flat case, we give a negative answer that there exists a 3dimensional closed Heisenberg infranilmanifold with cyclic holonomy of order bigger than or equal to 3, which cannot be diffeomorphic to a cusp cross-section of a 1-cusped finite volume complex hyperbolic 2-manifold.This is obtained from the formula by the characteristic numbers of bounded domains related to the Burns-Epstein invariant on strictly pseudoconvex CR-manifolds. McReynolds informed us that Neumann-Reid have obtained the similar answer using eta-invariant. D. B. McReynolds gave a necessary and sufficient condition for a Heisenberg infranilmanifold to be realized as a cusp-section of finite volume (arithmetically) complex hyperbolic orbifold (Algebr. and Geom. Topol. 2004 (4), 721755). We study this problem using the Seifert fibration and group extensions of Heisenberg infranilmanifold.

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Discrete Groups and Geometric Structures II On Marcus conjecture Inkang 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 of the identity component of the automorphism group of the universal cover of M , which is not Rn , is nonempty. A canonical compatible metric for geometric structures on nilmanifolds 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 left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the ‘invariant Ricci’ flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish two 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 are precisely the nilradicals of (standard) Einstein solvmanifolds. Our approach proposes to vary Lie brackets rather than inner products; our tool is the moment map for the action of a reductive Lie group on the variety of all Lie algebras, which we show to coincide in this setting with the Ricci operator. This gives us the possibility to use strong results from geometric invariant theory. Riemannian foliations on H2 × R Kyung Bai Lee University of Oklahoma joint work with S. Yi A foliation F on a Riemannian manifold M is said to be Riemannian (=metric) if the leaves of F are locally everywhere equidistant, and homogeneous if locally, its leaves coincide with the orbits of some group of isometries acting freely on M . As is well known, every homogeneous foliation is a metric foliation. The converse is known to be true on S 2 × R, Heisenberg group H2n+1 . For 1-dimensional foliations, the converse is also true for constant nonnegative-curved manifolds, while spaces of negative sectional curvature admit an 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) orbit of a subgroup of the group of isometries. The main result is

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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, because there exist at least two kinds of 1-dimensional metric foliations on X which are not homogeneous, (these do not contain any homogeneous leaves). Arithmetic cusp shapes are dense David McReynolds University of Texas Given a flat or almost flat manifold N , one can ask whether or not N can be smoothly embedded or isometrically embedded as a cusp cross-section of an arithmetic real, complex, or quaternionic hyperbolic orbifold. In previous work we answered the smooth embedding problem. In this talk, we discuss recent work on which metrics can be realized in cusp crosssections of arithmetic orbifolds. In particular, for flat manifolds, the flat metrics which can be realized in the cusp cross-sections of arithmetic real hyperbolic orbifolds are dense in the moduli space of flat metrics. Similar results hold for infra-nilmanifolds modelled on the Heisenberg group and its quaternionic analog provided a smooth embedding exists. Isometric actions of Heisenberg groups on compact Lorentz manifolds Karin Melnick University of Chicago Connected isometry groups of compact connected Lorentz manifolds have been classified by Adams and Stuck and independently by Zeghib. The three non-compact, non-abelian groups that can occur as direct factors are P SL2 (R), Heisenberg groups, and certain solvable extensions of Heisenberg groups. In the first and third cases, splitting theorems due to Gromov and Zeghib, respectively, tell what the manifolds with these isometry groups can be. We prove some results toward classifying compact Lorentz manifolds on which Heisenberg groups act isometrically. The main result is a classification of actions in which the dimension of the Heisenberg group is one less than the dimension of the manifold. Boundaries of systolic groups Damian Osajda University of Wroclaw Systolic complexes and groups (i.e. groups acting geometrically on them) were introduced by 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 nonpositive curvature. In particular they give rise to some new examples of hyperbolic (in the sense of Gromov) groups (although a class of them is different than the class of hyperbolic

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groups). We will show that boundaries of hyperbolic groups constructed this way are strongly hereditarily aspherical compacta of arbitrary dimension. This gives a new class of spaces that can occur as boundaries of hyperbolic groups (note that only few (and relatively small) classes of spaces are known to be like that). Moreover those boundaries are interesting themselves if one studies their topology (for those that we can describe any way). We will also show that studying topology ”far away” of systolic (maybe not hyperbolic) groups allows us to distinguish them among many other groups arising in geometric group theory. We plan to state some open problems and conjectures too. Endomorphisms of relatively hyperbolic groups Andrzej Szczepanski University of Gdansk joint work with I. Belegradek We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups and in particular prove the following. • If G is a finitely generated non-elementary relatively hyperbolic group with slender parabolic subgroups , and either G is not co-Hopfian our Out(G) is infinity, then G splits over a slender group. • If a finitely generated non-parabolic subgroup H of a non-elementary relatively hyperbolic 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 of Out(H) for some Kazhdan group H. (This sharpens a result of Oliver-Wise)

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

Abstracts of short communications

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Covolumes of Uniform Lattices acting on Hyperbolic Buildings 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 covolumes µ(Γ\G), for G an algebraic group. More recently, the question of uniform covolumes has been posed for G the automorphism group of a tree. We begin the investigation of the case where G is the automorphism group of a 2-dimensional simplicial complex, such as a hyperbolic building. In particular, when G is the automorphism group of Bourdon’s building Ipq , we find the exact set of covolumes of uniform lattices in G. Hyperbolic Coxeter groups Pavel Tumarkin Independent University of Moscow We consider discrete groups generated by reflections in the hyperbolic space. In contrast to 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 concerning groups with relatively small number of generating reflections. Non-injective representations of a closed surface group into P SL(2, R) Maxine Wolff Institut Fourier, Grenoble Let e denote the Euler class on the space Hom(Γg , P SL(2, R)) of representations of the group Γg of the cosed surface Σg of genus g. Goldman showed that the connected components of Hom(Γg , P SL(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 injective representations whose image is a discrete subgroup of P SL(2, R). We prove that nonfaithful representations are dense in all the other components. We show that the image of a discrete representation essentially determines its Euler class. Moreover, we show that for every genus and possible corresponding Euler class, there exist discrete representations.

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Poster session On Lorentz dynamics : from group actions to warped products. Abdelouahab Arouche, Alger USTHB We show a geometric rigidity of isometric actions of noncompact simple Lie groups on Lorentz manifolds. Namely, we show that the manifold has a local warped product structure of a Lorentz manifold with constant curvature by a Riemannian manifold. On the Limit Set of Discrete Subgroups of P U (2, 1) Juan Pablo Navarrete Carrillo It is well-known that the elements of P SL(2, C) are classified as elliptic, parabolic or loxodromic according to the dynamics and their fixed points; these three types are also distinguished by their trace. If we now look at the elements in P U (2, 1), then W. Goldman introduced the equivalent notions of elliptic, parabolic or loxodromic elements and classified them by their trace. In this work we extend Goldman’s classification to all elements of P SL(3, C); we also extend to this setting the theorem that classifies them according to their trace. We then use this classification to study and compare two different notions of the limit set of a discrete subgroup of P U (2, 1). The first of these is due to Chen and Greenberg, given by thinking of these groups as automorphisms of the complex hyperbolic 2-space; the second definition of the limit set is due to Kulkarni and comes from these automorphisms of the whole complex projective plane. Representations of symmetry groups of carbon nanotubes and applications Nicolae Cotfas University of Bucharest A single-wall carbon nanotube is a highly symmetric quasi-one-dimensional cylindrical structure, which can be visualized as the structure obtained by rolling a honeycomb lattice such that the endpoints of a translation vector are folded one onto the other. The symmetry group of nanotube depends on this vector and is one of the line groups Lqp 22, 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 using a system of generators for the symmetry group. Each atomic position corresponds to an element of the set Z × {1, 2, . . . , n} × {0, 1}, where n is a natural number depending on the considered nanotube. We obtain an alternate rather different description by starting from

Abstracts of posters

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a description of the honeycomb lattice in terms of Miller indices. In our mathematical model 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 mathematical objects with geometric of physical significance have a simpler and more symmetric form [3-6]. References. [1] M. Damnjanovi´ c et al., Symmetry and lattices of single-wall nanotubes, J. Phys. A: Math. Gen., 32(1999), 4097-4104. [2] M. Damnjanovi´ c et al., Wigner-Eckart theorem in the inductive spaces and applications to optical transitions in nanotubes, 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 motions Tine 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 translations. 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 G acting by pure translations. This result no longer holds in case the commutator subgroup is higher dimensional. We also determine all five-dimensional nilpotent Lie groups acting simply transitively and affinely on R5 in such a way that only the identity element acts as a 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 group is free 2-step nilpotent and in case the Lie group is filiform. The Anosov relation for Nielsen numbers of maps on infra-nilmanifolds Bram De Rock K.U.Leuven Campus Kortrijk joint work with K. Dekimpe and W. Malfait In fixed point theory, the Nielsen number N (f ) and the Lefschetz number L(f ) are two numbers associated with a continuous self-map f : M → M of a smooth closed manifold M to provide information on the fixed points of f . N (f ) gives more information, but unfortunately N (f ) is not readily computable from its definition while L(f ) is much easier

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to calculate. A celebrated theorem of Anosov states that for any continuous map f : M → M of a nilmanifold M , one has that N (f ) = |L(f )|. He also showed that this result does not hold in general for infra-nilmanifolds, since he constructed a counterexample on the Klein bottle. We show that Anosov’s theorem is still true for infra-nilmanifolds with odd order holonomy group. We also establish a necessary and sufficient condition for expanding maps f of infra-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 groups Anastasia Evangelidou Larnaka, Cyprus Let G be a group, t an unknown and r(t) an element of the free product G ∗ hti. The equation 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) is non-zero the equation has a solution. There have been several results concerning the KL conjecture when the problem is restricted 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, for example for equations of certain length. In particular it has been proved that the KL conjecture 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 permutation and inversion: r0 (t) = atbtctdtet = 1, r1 (t) = atbtctdtet−1 = 1 , r2 (t) = atbtctdt−1 et−1 = 1, r3 (t) = atbtct−1 dtet−1 = 1. The aim of this presentation is to illustrate the methodology used for the solution of the equation r(t) = 1 that uses curvature arguments on relative diagram and the weight test 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.

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Coxeter decompositions of polytopes Anna Felikson Independent University of Moscow Let P be a polytope in S n , E n or H n . P is called a Coxeter polytope if all dihedral angles of P are submultiples of π. A Coxeter decomposition of P is a tiling of P by finite number of Coxeter polytopes such that if two tiles of the decomposition have a common facet then these tiles are symmetric to each other with respect to this facet. Coxeter decomposition of a Coxeter polytope corresponds to a reflection subgroup of the reflection group generated by the reflections with respect to the facets of the polytope. Decompositions of non-Coxeter polytopes correspond to non-standard systems of generators for the same group. We discuss general properties of Coxeter decompositions and decompositions of the polytopes with low numbers of facets.

Groupes moyennables, dimension topologique moyenne et sous-decalages Fabrice Krieger IRMA, Strasbourg Un th´eor`eme dˆ u a` Jaworski affirme que toute action continue minimale d’un groupe commutatif G sur un espace compact m´etrisable X de dimension topologique finie, se plonge dans le G-d´ecalage sur [0, 1]G . Pour G = Z, E. Lindenstrauss et B. Weiss ont montr´e qu’on ne pouvait pas supprimer l’hypoth´ese de finitude de la dimension topologique de X en construisant un contre-exemple. Nous allons g´en´eraliser le r´esultat de Lindenstrauss-Weiss et d´emontrer que si G est un groupe infini d´enombrable, moyennable et r´esiduellement fini, alors il existe un espace compact m´etrisable X, muni d’une action continue minimale de G, tel que le syst`eme dynamique obtenu ne se plonge pas dans le G-d´ecalage sur [0, 1]G . Cela montre en particulier que l’on ne peut pas supprimer l’hypoth`ese de finitude de la dimension topologique de X dans le th´eor`eme de Jaworski lorsque G est de type fini. L’outil essentiel utilis´e dans la d´emonstration du r´esultat est la dimension topologique moyenne qui est un invariant topologique des actions de groupes moyennables introduit par M. Gromov. En utilisant cet invariant, on r´eduit le probl`eme a` la construction d’un G-syst´eme minimal X de dimension topologique moyenne plus grande que celle du G-d´ecalage [0, 1]G .

Commutative polarizations Alfons 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 of L. A necessary and sufficient condition is given in order for D(P ) to be a maximal subfield

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of D(L). This settles a question by Jacques Alev. This condition is satisfied if P is a commutative 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 demonstrate their widespread occurrence. In particular, we will look at the corresponding induced representations and their kernels, the primitive ideals of U (L). Special attention is devoted to the situation where L is a semi direct product S ⊕ P , where P is a commutative ideal of L. For instance, let k be algebraically closed and let S be a simple Lie algebra, acting irreducibly 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 L of 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 an explicit formula for the index i(N ) of N .

List of participants

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List of participants Herbert Abels Fakult¨at f¨ ur Mathematik, Universit¨at Bielefeld Postfach 100131, D-33501 Bielefeld, Germany [email protected] Kristina Altmann Technische Universit¨at Darmstadt Fachbereich Mathematik, AG5, Schlossgassenstr. 7, 64289 Darmstadt, Germany [email protected] Sergei Antonyan Universidad Nacional Autonoma de Mexico Dept. Mat., Facultad de Ciencias, UNAM, 04510 Mexico D.F., Mexico [email protected] Abdelouahab Arouche USTHB Alger BP 32 USTHB 16123 Bab-Ezzouar Alger, Algeria [email protected] Thierry Barbot CNRS / ENS Lyon ENS Lyon UMPA, 46 All´ee d’Italie, 69364 Lyon, France [email protected] Oliver Baues Universit¨at Karlsruhe Mathematisches Institut II, Englerstr. 2, 76128 Karlsruhe, Germany [email protected] Yves Benoist ENS Paris 45 Rue d’Ulm, 75005 Paris, France [email protected] Adalbert Bovdi University of Debrecen Institute of Mathematics and Informatics, University of Debrecen, P.O.Box 12, Debrecen H-4010, Hungary [email protected] Victor Bovdi University of Debrecen Institute of Mathematics and Informatics, University of Debrecen, P.O.Box 12, Debrecen H-4010, Hungary [email protected] Emmanuel Breuillard IHES 35 Route de Chartres, 91440 Bures-sur-Yvette, France [email protected]

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Martin Bridson Imperial College London Mathematics, Huxley Building, Imperial College London, London SW7 2AZ, England, UK [email protected] Dietrich Burde Universit¨at Wien Fak. Mathematik, Nordbergstr. 15, 1090 Wien, Austria [email protected] Pierre-Emmanuel Caprace Universit´e Libre de Bruxelles ULB CP216, Boulevard du Triomphe, 1050 Bruxelles, Belgium [email protected] Virginie Charette University of Manitoba Dept. of Mathematics, University of Manitoba, Winnipeg MB R3T2N2, Canada [email protected] Suhyong Choi KAIST Daejeon, Yuseong-Gu Guseong 305-701, South Korea [email protected] Nicolae Cotfas University of Bucharest P.O. Box 76-54, Post Office 76, Bucharest, Romania [email protected] Pallavi Dani University of Chicago 5135 S. University Avenue, #3B, Chicago, IL 60615, United States of America [email protected] Tine De Cat K.U.Leuven campus Kortrijk Etienne Sabbelaan 53, B-8500 Kortrijk, Belgium [email protected] Yves de Cornulier EPF Lausanne EPFL, Dept. de Math´ematiques, Dat. MA, CH1015 Lausanne, France [email protected] Karel Dekimpe K.U.Leuven campus Kortrijk Etienne Sabbelaan 53, B-8500 Kortrijk, Belgium [email protected] Pierre de la Harpe Universit´e de Gen`eve Section de Math´ematiques, CP64, 1211 Gen`eve 4, Switserland [email protected]

List of participants

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Bram De Rock K.U.Leuven campus Kortrijk Etienne Sabbelaan 53, B-8500 Kortrijk, Belgium [email protected] Sandra Deschamps K.U.Leuven campus Kortrijk Etienne Sabbelaan 53, B-8500 Kortrijk, Belgium [email protected] Franki Dillen K.U.Leuven Celestijnenlaan 200B, B-3001 Heverlee, Belgium [email protected] Anastasia Evangelidou Evripidou 39, Larnaka 6036, Cyprus [email protected] Benson Farb University of Chicago 5734 S. University Avenue, Chicago IL 60637, United States of America [email protected] Yves F´ elix Universit´e Catholique de Louvain Bˆatiment Marc de Hemptinne, Chemin du Cyclotron 2, B-1348 Louvain-la-Neuve, Belgium [email protected] Anna Felikson MPIM Bonn / Independent University of Moscow MPI for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany [email protected] Oscar Garcia-Prada CSIC Madrid Instituto de Matem´aticas y F´ısica Fundamental, Consejo Superior de Investigaciones Cient´ıficas, Serrano, 113 bis, 28006 - Madrid, Spain [email protected] Etienne Ghys ENS Lyon ENS Lyon UMPA, 46 All´ee d’Italie, 69364 Lyon, France [email protected] William Goldman University of Maryland University of Maryland, College Park, MD 20742, United States of America [email protected] Fritz Grunewald Universit¨at Du¨esseldorf Mathematisches Institut, Heinrich Heine Universit¨at D¨ usseldorf, Germany [email protected]

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Mohammed Guediri King Saud University Dept. of Mathematics, College of Science, King Saud University, P.O.Box 2455, Riyadh 11451, Saudi Arabia [email protected] Antonin Guilloux Ecole Normale Sup´erieure 45 Rue d’Ulm, 75005 Paris, France [email protected] Toshiaki Hattori Tokyo Institute of Technology Dept. of Mathematics, Oh-okayama, Meguro, Tokyo 152-8551, Japan [email protected] Paul Igodt K.U.Leuven campus Kortrijk Etienne Sabbelaan 53, B-8500 Kortrijk, Belgium [email protected] Delarem Kahrobaei University of St. Andrews Mathematical Institute (Room 315), University of St. Andrews, North Haugh, St. Andrews, Fife, KY 16 9SS, Scotland, UK [email protected] Yoshinobu Kamishima Tokyo Metropolitan University Dept. of Mathematics, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji, Tokyo 192-0397 , Japan [email protected] Caroline Keil Universit¨at D¨ usseldorf Mathematisches Institut, Heinrich Heine Universit¨at D¨ usseldorf, Germany [email protected] Inkang Kim Seoul National University Seoul National University, Math. Dept., 151-342 Seoul, South Korea [email protected] Ji-Ae Kim Seoul National University Seoul National University, Math. Dept., 151-342 Seoul, South Korea [email protected] Benjamin Klopsch Universit¨at Du¨esseldorf Mathematisches Institut, Heinrich Heine Universit¨at D¨ usseldorf, Germany [email protected] Fabrice Krieger IRMA Strasbourg IRMA, UMR 7501 CNRS/ULP, 7 rue Ren´e Descartes, 67084 Strasbourg Cedex, France [email protected]

List of participants

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Jorge Lauret FAMAF, Universidad N´acional de C´ordoba Haya de la Torre s/n, 5000 C´ordoba, Argentina [email protected] Kyung Bai Lee University of Oklahoma Department of Mathematics, University of Oklahoma, Norman, OK 73019, United States of America kb [email protected] Alexander Lichtman University of Wisconsin-Parkside Kenosha, WIS314, United States of America [email protected] Seonhee Lim ENS Paris / Yale University 45 Rue d’Ulm, 75005 Paris, France [email protected] Gabriele Link Universit¨at Karlsruhe Mathematisches Institut II, Englerstr. 2, 76128 Karlsruhe, Germany [email protected] Isabelle Liousse Universit´e de Lille 1 Laboratoire Painlev´e, Universit´e de Lille 1, 59655 Villeneuve d’ascq cedex, France [email protected] Wim Malfait K.U.Leuven campus Kortrijk Etienne Sabbelaan 53, B-8500 Kortrijk, Belgium [email protected] Alec Mason Glasgow University Dept. of Mathematics, Glasgow G12 P9W, Scotland, UK [email protected] Ben McReynolds University of Texas 4208 Avenue B, Austin TX 78751, United States of America [email protected] Karin Melnick University of Chicago 5734 S. University Avenue, Chicago IL 60637, United States of America [email protected] Nicolas Michelacakis University of Cyprus Mathematics and Statistics Department, University of Cyprus, P.O. Box 20537, Nicosia 1678, Cyprus [email protected]

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Igor Mineyev University of Illinois at Urbana-Champaign Dept. of Mathematics, UIUC, Urbana IL61801, United States of America [email protected] Bernhard M¨ uhlherr Universit´e Libre de Bruxelles ULB CP216, Boulevard du Triomphe, 1050 Bruxelles, Belgium [email protected] Juan Pablo Navarrete Carrillo Universidad Nacional Autonoma de Mexico Dept. Mat., Facultad de Ciencias, UNAM, 04510 Mexico D.F., Mexico [email protected] Guennadi Noskov Fakult¨at f¨ ur Mathematik, Universit¨at Bielefeld Postfach 100131, D-33501 Bielefeld, Germany [email protected] Alfons Ooms Limburgs Universitair Centrum Dept. WNI E87, Limburgs Universitair Centrum, Universitaire Campus, B-3590 Diepenbeek, Belgium [email protected] Damian Osajda University of Wroclaw Kielczowska 51b/2, 51-315 Wroclaw, Poland [email protected] John Panagopoulos University of Athens University of Athens, Panepistimiopolis, Athens 15784, Greece [email protected] Krzystof Pawalowski Adam Mickiewick University Ul. Umultowska 87, 61614 Pozn´an, Poland [email protected] Hannes Pouseele K.U.Leuven campus Kortrijk Etienne Sabbelaan 53, B-8500 Kortrijk, Belgium [email protected] Bartosz Putrycz University of Gdansk Instytut Matematyki UG, ul. Wita Stwosza 57, 80-952 Gdansk, Poland [email protected] Andrzei Szczepanski University of Gdansk Instytut Matematyki UG, ul. Wita Stwosza 57, 80-952 Gdansk, Poland [email protected]

List of participants Olympia Talelli University of Athens University of Athens, Panepistimiopolis, Athens 15784, Greece [email protected] Anne Thomas University of Chicago 5734 S. University Avenue, Chicago IL 60637, United States of America [email protected] Paulo Tirao FAMAF, Universidad N´acional de C´ordoba Haya de la Torre s/n, 5000 C´ordoba, Argentina [email protected] Domingo Toledo IHES 35 Route de Chartres, 91440 Bures-sur-Yvette, France [email protected] Pavel Tumarkin MPIM Bonn MPI for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany [email protected] Joost Van Hamel K.U.Leuven Celestijnenlaan 200B, B-3001 Heverlee, Belgium [email protected] Stephen Wang University of Chicago 5510 S. Wood Lawn #402, United States of America [email protected] Cynthia Will FAMAF, Universidad N´acional de C´ordoba Haya de la Torre s/n, 5000 C´ordoba, Argentina [email protected] Maxine Wolff Institut Fourier, Grenoble 100, rue des Maths, BP74 38402 St Martin d’Heres, France [email protected]

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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 and Techniques in Mathematics • the Universit´ e Catholique de Louvain (UCL), and • the Katholieke Universiteit Leuven (Campus Kortrijk) for their support.