Small doubling for discrete subsets of non-commutative groups and a theorem of Lagarias

We prove the stable rationality of almost simple adjoint algebraic groups, the connected components of the Dynkin diagram of anisotropic kernel of which contain at most two vertices. The (stable) rationality of many isotropic almost simple groups with small anisotropic kernel and some related results over arbitrary fields are discussed.

Almost independence and irreducibility in simple finite and algebraic groups

On the Tits indices of absolutely almost simple algebraic groups over local and global fields

Products of conjugacy classes in finite and algebraic simple groups

Invariant forms on irreducible modules of simple algebraic groups

Let G be a simple classical algebraic group over an algebraically closed field K of characteristic p ? 0 with natural module W. Let H be a closed subgroup of G and let V be a nontrivial irreducible tensor indecomposable -restricted rational KG-module such that the restriction of V to H is irreducible. In this paper we classify the triples (G,H,V ) of this form, where H is a closed disconnected almost simple positive-dimensional subgroup of G acting irreducibly on W. Moreover, by combining this result with earlier work, we complete the classifcation of the irreducible triples (G,H,V ) where G is a simple algebraic group over K, and H is a maximal closed subgroup of positive dimension.

We continue the investigation of infinite, definably simple groups which are definable in o-minimal structures. InDefinably simple groups in o-minimal structures, we showed that every such group is a semialgebraic group over a real closed field. Our main result here, stated in a model theoretic language, is that every such group is either bi-interpretable with an algebraically closed field of characteristic zero (when the group is stable) or with a real closed field (when the group is unstable). It follows that every abstract isomorphism between two unstable groups as above is a composition of a semialgebraic map with a field isomorphism. We discuss connections to theorems of Freudenthal, Borel-Tits and Weisfeiler on automorphisms of real Lie groups and simple algebraic groups over real closed fields.

We complete the study of characters on higher rank semisimple lattices initiated in [BH21, BBHP22], the missing case being the case of lattices in higher rank simple algebraic groups in arbitrary characteristics. More precisely, we investigate dynamical properties of the conjugation action of such lattices on their space of positive definite functions. Our main results deal with the existence and the classification of characters from which we derive applications to topological dynamics, ergodic theory, unitary representations and operator algebras. Our key theorem is an extension of the noncommutative Nevo–Zimmer structure theorem obtained in [BH21] to the case of simple algebraic groups defined over arbitrary local fields. We also deduce a noncommutative analogue of Margulis’ factor theorem for von Neumann subalgebras of the noncommutative Poisson boundary of higher rank arithmetic groups.

This note describes a unified approach to several superrigidity results, old and new, concerning representations of lattices into simple algebraic groups over local fields. For an arbitrary group $\Gamma$ and a boundary action $\Gamma$ ↷ $B$ we associate a certain generalized Weyl group $W_{{\Gamma}{B}}$ and show that any representation with a Zariski dense unbounded image in a simple algebraic group, $\rho:\Gamma\to \bf{H}$, defines a special homomorphism $W_{{\Gamma}{B}}\to Weyl_{\bf H}$. This general fact allows the deduction of the aforementioned superrigidity results.

LetGbe a simple adjoint algebraic group over an algebraically closed fieldK. We are concerned to describe the conjugacy classes of unipotent elements ofG. Goperates on its Lie algebra g by means of the adjoint action and we may consider classes of nilpotent elements of g under this action. It has been shown by Springer (11) that there is a bijection between the unipotent elements ofGand the nilpotent elements ofgwhich preserves theG-action, provided that the characteristic ofKis either 0 or a ‘good prime’ forG. Thus we may concentrate on the problem of classifying the nilpotent elements of g under the adjointG-action.

We introduce property (T_\mathrm {Schur},G,K) and prove it for some non-cocompact lattice in Sp_4 in a local field of finite characteristic. We show that property (T_\mathrm {Schur},G,K) for a non-cocompact lattice \Gamma in a higher rank almost simple algebraic group in a local field is an obstacle to proving the Baum Connes conjecture without coefficient for \Gamma with known methods, and this is stronger than the well-known fact that \Gamma does not have the property of rapid decay (property (RD)). It is the first example (as announced in [7]) for which all known methods for proving the Baum–Connes conjecture without coefficient fail.

In the last twenty years there has been a tremendous amount of progress in our understanding of subgroup structure of simple groups G, both algebraic over an algebraically closed field and finite. If G is of exceptional type, the progress has been particularly impressive, largely due to the work of Liebeck and Seitz; an excellent survey is the article [25] of Liebeck in this volume. If G is a finite classical simple group, the reduction theorem of Aschbacher [1] enables us to concentrate on the case where the subgroup H is almost simple modulo the subgroup Z of scalars and the (projective) representation of the simple group F* (H/Z) on the natural module V for G is absolutely irreducible. There is a similar reduction theorem for G a classical algebraic group over an algebraically closed field [28]. There are a number of survey articles where the situation is discussed — see, e.g., [21], [24], [35], and [39].

We describe the global structure of totally disconnected locally compact groups having a linear open compact subgroup. Among the applications, we show that if a non-discrete, compactly generated, topologically simple, totally disconnected locally compact group is locally linear, then it is a simple algebraic group over a local field.

This book concerns the theory of unipotent elements in simple algebraic groups over algebraically closed or finite fields, and nilpotent elements in the corresponding simple Lie algebras. These topics have been an important area of study for decades, with applications to representation theory, character theory, the subgroup structure of algebraic groups and finite groups, and the classification of the finite simple groups. The main focus is on obtaining full information on class representatives and centralizers of unipotent and nilpotent elements. Although there is a substantial literature on this topic, this book is the first single source where such information is presented completely in all characteristics. In addition, many of the results are new--for example, those concerning centralizers of nilpotent elements in small characteristics. Indeed, the whole approach, while using some ideas from the literature, is novel, and yields many new general and specific facts concerning the structure and embeddings of centralizers.

It was the aim of the meeting to bring together international experts from the theory of buildings, differential geometry and geometric group theory. Buildings are combinatorial structures (simplicial complexes) which can be seen as simultaneously generalizing projective spaces and trees. Already from these examples it is clear that there will be interesting groups acting on buildings. Conversely, groups can be studied using their actions on given buildings. Groups coming up in this context are in particular groups having a BN -pair. Examples of such groups include the classical groups, simple Lie groups and algebraic groups (also over local fields), Kac-Moody groups and loop groups. This already indicates that these groups play an important role in many different areas of mathematics such as algebra, geometry, number theory, physics and analysis. Kac-Moody groups correspond to so-called twin buildings, a particularly active area in the theory of buildings. Geometric group theory is concerned with the investigation of group actions on metric spaces using the interplay of group theoretic properties and metric properties like curvature in the sense of Alexandrov, or CAT (0) -spaces. The geometric realization of a building is a metric space with interesting curvature properties on which the above mentioned groups as well as their subgroups like uniform lattices or arithmetic groups act in a natural way by isometries. In this respect there are a number of canonical connections between the theory of buildings and geometric group theory. One of the current problems concerns the characterization of buildings as metric spaces. In differential geometry these aspects also play an important role, e.g. in connection with Hadamard manifolds, (simply connected Riemannian manifolds of nonpositive curvature). A special role is played by the Riemannian symmetric spaces and their quotients of finite volume which one wants to characterize geometrically. By considering the fundamental groups, one obtains discrete group actions also studied in geometric group theory. Buildings come up in differential geometry as the compactifications of Riemannian symmetric spaces yielding examples of topological buildings. Asymptotic cones (and ultrapowers) of symmetric spaces present non-discrete affine buildings and create new and interesting relations to model theory. These constructions are important in new proofs of differential geometric rigidity theorems, like Mostow Rigidity and the Margulis Conjecture. This shows that there are close connections between the areas, and this meeting was the first in a number of years in Oberwolfach having these connections as its topic. Geometric group theory has recently introduced interesting aspects into the theory of buildings, in particular the hyperbolic buildings. Conversely, new developments in the theory of buildings, e.g. the twin buildings have interesting group theoretic applications, for example in the theory of S -arithmetic groups or in the theory of Kac-Moody groups. All these aspects played an important part in this meeting and the interaction between the participants from different areas was very lively.