Bertrand Rémy

Bruhat-Tits buildings and analytic geometry

This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry can be used to compactify buildings. We discuss in detail the example of the special linear group.

Symmetries, integrable systems and representations

A presentation of the deformed W1+1 algebra.- Generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes.- PBW filtration over Z and compatible bases for VZ(_) in type An and Cn.- On the subgeneric restricted blocks of affine category O at the critical level.- Slavnov determinants, Yang-Mills structure constants, and discrete KP.- Monodromy of partial KZ functors for rational Cherednik algebras.- Category of finite dimensional modules over an orthosymplectic Lie superalgebra: small rank examples.- Monoidal categorifications of cluster algebras of type A and D.- A classification of roots of symmetric Kac-Moody root systems and its application.- Fermions acting on quasi-local operators in the XXZ model.- The Romance of the Ising Model.- A(1) n -Geometric Crystal corresponding to Dynkin index i = 2 and its ultra-discretization.- A Z3-orbifold theory of lattice vertex operator algebra and Z3-orbifold constructions.- Words, automata and Lie theory for tilings.- Toward Berenstein-Zelevinsky data in affine type A, part III: Proof of the connectedness.- Quiver varieties and tensor products, II.- Derivatives of Schur, Tau and Sigma Functions on Abel-Jacobi Images.- Pade interpolation for elliptic Painleve equation.- Non-commutative harmonic oscillators.- The inversion formula of polylogarithms and the Riemann-Hilbert problem.- Some remarks on the Quantum Hall Effect.- Ordinary differential equations on rational elliptic surfaces.- On the spectral gap of the Kac walk and other binary collision processes on d-dimensional lattice.- A restricted sum formula for a q-analogue of multiple zeta values.- A trinity of the Borcherds _-function.- Sum rule for the eight-vertex model on its combinatorial line.

Geometric Structures in Group Theory

The overall theme of the conference was geometric group theory, interpreted quite broadly. In general, geometric group theory seeks to understand algebraic properties of groups by studying their actions on spaces with various topological and geometric properties; in particular these spaces must have enough structure-preserving symmetry to admit interesting group actions. Although traditionally geometric group theorists have focused on finitely generated (and even finitely presented) countable discrete groups, the techniques that have been developed are now applied to more general groups, such as Lie groups and Kac-Moody groups, and although metric properties of the spaces have played a key role in geometric group theory, other structure such as complex or projective structures and measure-theoretic structures are being used more and more frequently. Mathematics Subject Classification (2010): 20Fxx, 57Mxx. Introduction by the Organisers In addition to discussing the most recent developments within geometric group theory, the meeting also highlighted several dramatic contributions of geometric group theory to other fields. A particular emphasis within the field was studying several classes of groups which exhibit properties of classical examples such as arithmetic groups but are not themselves arithmetic. The idea that a group can be thought of as a geometric object with non-positive or negative curvature is one of the most fundamental ideas in geometric group theory. Curvature conditions have helped us to understand both the general, randomly defined group and specific families of groups arising from topological of 1630 Oberwolfach Report 28/2013 differential-geometric considerations. The focus has recently shifted to variants on these curvature conditions, both those which were defined long ago but not intensively studied and newly introduced notions. For example, Gromov introduced “relative hyperbolicity” at the same time as he defined hyperbolicity, but this was not studied deeply until at least a decade later. Relative hyperbolicity captures behavior similar to that of non-uniform lattices in real hyperbolic spaces in a more general, non-smooth framework. Other variants of hyperbolicity focus on properties of a particular group action rather on the group itself, and generalize classical small cancellation theory. This has led to the construction of quotient groups with prescribed properties, starting from a suitable action of a group on a space, and has had applications to groups arising from unexpected quarters, such as proving that the Cremona group is not simple. Several talks during the week dealt with new techniques and questions. For example, in some talks the use of an auxiliary space with a group action is less central, such as in investigating the possible growth rates of finitely generated groups, or in attempts to establish a general theory of totally disconnected locally compact groups. In others, the structures on spaces preserved by the group action are more of an analytic nature than a geometric one, for example there are some exciting connections with measure theory and with operator algebras, some of which lead to deep topological questions. Specific families of groups that were considered in the talks included mapping class groups MCG(Σ) of surfaces, groups of outer automorphisms Out(Fn) of nonabelian free groups and isometry groups of buildings. These are of particular interest because of their connections with many other areas of mathematics, and because each in its way generalizes the classical examples of linear groups acting on symmetric spaces. The construction of suitable substitutes for the symmetric spaces and the investigation of even the most basic properties are often very difficult. Certainly one of the most exciting developments in the field was the recent use of geometric group theory to solve the last open conjecture on W. Thurston’s famous list of problems on the structure of 3-manifolds. Two speakers gave talks explaining both the geometric group theory and its application to 3-manifolds during the official schedule, and informal sessions were held in the evenings for those who wanted to hear more details. Progress is currently being made on simplifying some of the proofs, and there are many further potential applications of the technology to geometric group theory. We had 52 participants from a wide range of countries, and 26 official lectures. The staff in Oberwolfach was—as always—extremely supportive and helpful. We are very grateful for the additional funding for 5 young PhD students and recent postdocs through Oberwolfach-Leibniz-Fellowships. In addition, there was one young student funded through the DMV Student’s Conference. We think that this provided a great opportunity for these students. We feel that the meeting was exciting and highly successful. The quality of all lectures was outstanding, and outside of lectures there was a constant buzz Geometric Structures in Group Theory 1631 of intense mathematical conversations. We are confident that this conference will lead to both new and exciting mathematical results and to new collaborations. Geometric Structures in Group Theory 1633 Workshop: Geometric Structures in Group Theory

Non-positive Curvature and Geometric Structures in Group Theory

The focus of this meeting was the use of geometric methods to study infinite discrete groups. Key topics included isometric actions of such groups on spaces of nonpositive curvature, such as CAT(0) cube complexes, buildings, and hyperbolic or symmetric spaces. These actions lead to a rich and fruitful interplay between geometry and group theoretic questions. Mathematics Subject Classification (2000): 20Fxx, 57Mxx. Introduction by the Organisers The meeting focused on several areas of current excitement in geometric group theory, unified by the important role that non-positive curvature plays in each. The geometric approach to group theory dominates the modern study of finitely generated groups. A central idea in this approach is that one illuminates the nature of groups by studying their actions on spaces with appropriate geometric structure. The quality of information one gleans from the action depends on the richness of the geometric structure and the quality of the action (discrete and cocompact by isometries being the most desirable). A powerful illustration of this is provided by the study of isometric actions on spaces of non-positive curvature. The curvature hypothesis alone tells one a great deal about the algebraic structure of the group, but the theory becomes much richer when one imposes further hypotheses on the space. Prime illustrations of this are the the theory of buildings (J. Tits) and, most classically, the actions of discrete subgroups of semi-simple Lie groups on Riemannian symmetric spaces (É. Cartan). 1166 Oberwolfach Report 20/2010 The topics covered during this workshop can each be seen as a natural extension of an aspect of this last beautiful subject: rigidity, fixed point theorems, questions of linearity and residual finiteness, analysis at infinity, cohomological issues, etc. The diverse techniques involved in the topics that emerge under these headings typically lie far from these classical origins, and the spaces that arise are typically highly singular — buildings, CAT(0) cube complexes, asymptotic cones, the curve complex and other spaces related to Teichmüller space, Outer Space, etc. But the classical situation still provides a stimulating analogy. This diversity within a common framework was widely reflected in the speakers of the workshop. We concentrated on specific topics that have seen recent exciting progress. These include: the study of new classes of buildings, of CAT(0) cube complexes, lattices in the isometry groups of the latter spaces and related embedding results; recent insights into the nature of mapping class groups of surfaces and automorphism groups of free groups; recent definitive results on the nature of the full isometry groups of CAT(0) spaces that admit parabolics; and the introduction of powerful new tools of an analytic nature. More details can be seen in the individual abstract below. We had 55 participants from a wide range of countries, and 23 lectures. In addition, there were two special sessions in the evening, with lectures by Arthur Bartels on the recent proof of the Farrell-Jones Conjecture for hyperbolic and CAT(0) groups and by Mark Sapir on conjugacy growth in groups. The staff in Oberwolfach was—as always—extremely supportive and helpful. We are very grateful for the additional funding for five young PhD students and recent postdocs through Oberwolfach-Leibniz-Fellowships. In addition, there was one young student funded through the DMV Student’s Conference. We think that this provided a great opportunity for these students. We feel that the meeting was exciting and highly successful. The quality of all lectures was outstanding, and outside of lectures there was a constant buzz of intense mathematical conversations. We are confident that this conference will lead to both new and exciting mathematical results and to new collaborations.