Publication Areas

We develop a generalization of conditional probability for arbitrary ordered vector spaces. A related problem is that of assigning a numerical value to one vector relative to another.

We characterize the groups for which these generalized probabilities can be stationary, respectively invariant. Our results deviate from the setting of classical probability and lead to a new criterion for amenability and for fixed points in cones.

We establish an eigenfunctional theorem for positive operators, evocative of the Krein–Rutman theorem. A more general version gives a joint eigenfunctional for commuting operators.

We establish a fixed-point theorem for the face maps that consist in deleting the i-th entry of an ordered set. Furthermore, we show that there exists random finite sets of integers that are almost invariant under such deletions.  Consequences for various monoids of order-preserving transformations of N are discussed in an appendix.

We consider groups of piecewise-projective homeomorphisms of the line which are known to be non-amenable using notably the Carrière–Ghys theorem on ergodic equivalence relations. Replacing that theorem by an explicit fixed-point argument, we can strengthen the conclusion and exhibit uncountably many "amenability gaps" between various piecewise-projective groups.

We propose a fixed-point property for group actions on cones in topological vector spaces. In the special case of equicontinuous actions, we prove that this property always holds; this statement extends the classical Ryll-Nardzewski theorem for Banach spaces. When restricting to cones that are locally compact in the weak topology, we prove that the property holds for all distal actions, thus extending the general Ryll-Nardzewski theorem for all locally convex spaces.

Returning to arbitrary actions, the proposed fixed-point property becomes a group property, considerably stronger than amenability. Equivalent formulations are established and a number of closure properties are proved for the class of groups with the fixed-point property for cones. 

Warning: the formula in Definition 32 should be:  (1-ε) ‖ u⋅ f ‖  <  ‖(su)⋅ f ‖   <  (1+ε) ‖ u⋅ f ‖.

The corresponding correction has to be made for Problem 47. 

Consider the following property of a topological group G: every continuous affine G-action on a Hilbert space with a bounded orbit has a fixed point. We prove that this property characterizes amenability for locally compact σ-compact groups (e.g.  countable groups).

Along the way, we introduce a "moderate" variant of the classical induction of representations and we generalize the Gaboriau–Lyons theorem to prove that any non-amenable locally compact group admits a probabilistic variant of discrete free subgroups. This leads to the "measure-theoretic solution" to the von Neumann problem for locally compact groups.

We illustrate the latter result by giving a partial answer to the Dixmier problem for locally compact groups.

Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group IET of interval exchange transformations that have angular components of rational rank ≤ 2.

In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on IET and show that there are subgroups G < IET admitting no finitely supported measure with trivial boundary. 

We introduce a relative fixed point property for subgroups of a locally compact group, which we call relative amenability. It is a priori weaker than amenability. We establish equivalent conditions, related among others to a problem studied by Reiter in 1968. We record a solution to Reiter's problem.

We then study the class X of groups in which relative amenability is equivalent to amenability for all closed subgroups; we prove that X contains all familiar groups. Actually, no group is known to lie outside X.

Since relative amenability is closed under Chabauty limits, it follows that any Chabauty limit of amenable subgroups remains amenable if the ambient group belongs to the vast class X. 

The group of piecewise projective homeomorphisms of the line provides straightforward torsion-free counterexamples to the so-called von Neumann conjecture. The examples are so simple that many additional properties can be established.

Supramenability of groups is characterised in terms of invariant measures on locally compact spaces. This opens the door to constructing interesting crossed product C*-algebras for non-supramenable groups. In particular, stable Kirchberg algebras in the UCT class are constructed using crossed products for both amenable and non-amenable groups.

We provide the first examples of finitely generated simple groups that are amenable (and infinite). This follows from a general existence result on invariant states for piecewise-translations of the integers. The states are obtained by constructing a suitable family of densities on the classical Bernoulli space. 

Consider a proper cocompact CAT(0) space X. We give a complete algebraic characterisation of amenable groups of isometries of X. For amenable discrete subgroups, an even narrower description is derived, implying Q-linearity in the torsion-free case.

We establish Levi decompositions for stabilisers of points at infinity of X, generalising the case of linear algebraic groups to Isom(X). A geometric counterpart of this sheds light on the refined bordification of X (à la Karpelevich) and leads to a converse to the Adams–Ballmann theorem. It is further deduced that unimodular cocompact groups cannot fix any point at infinity except in the Euclidean factor; this fact is needed for the study of CAT(0) lattices.

Various fixed point results are derived as illustrations. 

We prove a fixed point theorem for a family of Banach spaces, notably for L1. Applications include the optimal answer to the "derivation problem" for group algebras which originated in the 1960s.

A simple characterisation of topological amenability in terms of bounded cohomology is proved, following Johnson's formulation of amenability. The connection to injective Banach modules is established.

We study property (T) and the fixed point property for actions on Lp and other Banach spaces. We show that property (T) holds when L2 is replaced by Lp (and even a subspace/quotient of Lp), and that in fact it is independent of  1 ≤ p < ∞. We show that the fixed point property for Lp follows from property (T) when  1 < p <  2+ε. For simple Lie groups and their lattices, we prove that the fixed point property for Lp holds for any  1 < p < ∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive Banach spaces.

We investigate the class of groups admitting an action on a set with an invariant mean. It turns out that many free products admit such an action. We give a complete characterisation of such free products in terms of a strong fixed point property.

We first show that co-amenability does not pass to subgroups, answering a question asked by Eymard in 1972. We then address co-amenability for von Neumann algebras, describing notably how it relates to the former.

For all algebraic groups over non-Archimedean local fields, the bounded cohomology vanishes. This follows from the corresponding statement for automorphism groups of Bruhat–Tits buildings, which hinges on the solution to the flatmate conjecture raised in earlier work with Bucher. Vanishing and invariance theorems for arithmetic groups are derived. 

We prove that the groups of orientation-preserving homeomorphisms and diffeomorphisms of Rn are boundedly acyclic, in all regularities. This is the first full computation of the bounded cohomology of a transformation group that is not compactly supported, and it implies that many characteristic classes of flat Rn- and Sn-bundles are unbounded. We obtain the same result for the group of homeomorphisms of the disc that restrict to the identity on the boundary, and for the homeomorphism group of the non-compact Cantor set.

In the appendix, Alexander Kupers proves a controlled version of the annulus theorem which we use to study the bounded cohomology of the homeomorphism group of the discs. 

It is, by now, classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list, namely uniformly stability with respect to the family of unitary operators on finite-dimensional Hilbert spaces equipped with submultiplicative norms. Towards this goal, we first build an elaborate coho- mological theory capturing the obstruction to such stability, and show that the vanishing of second cohomology implies uniform stability in this setting. This cohomology can be roughly thought of as an asymptotic version of bounded cohomology, and sheds light on a question raised in [Mon06] about a possible connection between vanishing of second bounded cohomology and Ulam stability. Along the way, we use this criterion to provide a short conceptual (re)proof of the classical result of Kazhdan [Kaz82] that discrete amenable groups are Ulam stable. We then use this machinery to establish our main result, that lattices in a class of higher rank semisimple groups (which are known to have vanishing bounded cohomology) are uniformly stable. 

We determine the bounded cohomology of the group of homeomorphisms of certain low-dimensional manifolds. In particular, for the group of orientation-preserving homeomorphisms of the circle and of the closed 2-disc, it is isomorphic to the polynomial ring generated by the bounded Euler class. These seem to be the first examples of groups for which the entire bounded cohomology can be described without being trivial.

We further prove that, contrary to ordinary cohomology, the diffeomorphisms groups of the circle and of the closed 2-disc have the same bounded cohomology as their homeomorphism groups and that both differ from the ordinary cohomology.

Finally, we determine the low-dimensional bounded cohomology of homeo- and diffeomorphism of the spheres Sn and of certain 3-manifolds. In particular, we answer a question of Ghys by showing that the Euler class in H2(Homeoo(S3)) is unbounded. 

We prove the vanishing of the bounded cohomology of lamplighter groups for a wide range of coefficients. This implies the same vanishing for a number of groups with self-similarity properties, such as Thompson's group F. In particular, these groups are boundedly acyclic. Our method is ergodic and applies to "large" transformation groups where the Mather–Matsumoto–Morita method sometimes fails because not all are acyclic in the usual sense.

We prove that the cohomology of semi-simple Lie groups admits boundary values, which are measurable cocycles on the Furstenberg boundary. This generalises known invariants such as the Maslov index on Shilov boundaries, the Euler class on projective space, or the hyperbolic ideal volume on spheres.

In rank one, this leads to an isomorphism between the cohomology of the group and of this boundary model. In higher rank, additional classes appear, which we determine completely. 

We prove the vanishing of the cup product of the bounded cohomology classes associated to any two Brooks quasimorphisms on the free group. This is a consequence of the vanishing of the square of a universal class for tree automorphism groups.

It is proved that the continuous bounded cohomology of SL2(k) vanishes in all positive degrees whenever k is a non-Archimedean local field. This holds more generally for boundary-transitive groups of tree automorphisms and implies low degree vanishing for SL2 over S-integers.

We prove that the norm of the Euler class E for flat vector bundles is 2–n (in even dimension n, since it vanishes in odd dimension). This shows that the Sullivan–Smillie bound considered by Gromov and Ivanov–Turaev is sharp. In the course of the proof, we construct a new cocycle representing E and taking only the two values ±2–n. Furthermore, we establish the uniqueness of a canonical bounded Euler class.

We prove vanishing results for Lie groups and algebraic groups (over any local field) in bounded cohomology. The main result is a vanishing below twice the rank for semi-simple groups. Related rigidity results are established for S-arithmetic groups and groups over global fields. We also establish vanishing and cohomological rigidity results for products of general locally compact groups and their lattices.

We establish the vanishing for non-trivial unitary representations of the bounded cohomology of SLd up to degree d-1. It holds more generally for uniformly bounded representations on superreflexive spaces. The same results are obtained for lattices. We also prove that the real bounded cohomology of any lattice is invariant in the same range.

A selection of aspects of the theory of bounded cohomology is presented. The emphasis is on questions motivating the use of that theory as well as on some concrete issues sugested by its study. Specific topics include rigidity, bounds on characteristic classes, quasification, orbit equivalence, amenability.

We prove that for all local fields SLn is stable over n in terms of continuous bounded cohomology. We complement this by various computations in low degree, showing notably the vanishing of the third bounded cohomology of SLn(R) for all n. We link the corresponding vanishing for p-adic fields to a question on prime numbers.

We establish how the spectral decomposition for a Riemann surface determines the allocation of the bounded cohomology over the representations of SL2(R). Then we explore the connections of the Dilogarithm with the continuous bounded cohomology of SL2(R) and SL2(C). In particular, it appears that Rogers' Dilogarithm is uniquely determined even measurably by the Spence–Abel functional equation.

The central theme of this paper is a product formula for (continuous) bounded cohomology, and more specifically its applications to rigidity theory for lattices — both in Lie/algebraic groups and more general topological groups. A more condensed exposition of some of the material published in the Lecture Note of the second author is followed by finiteness results for lattices. An appendix by Burger–Iozzi pins down a powerful use of boundary maps in this context.

The purpose of this monograph is (a) to lay the foundations for a conceptual approach to bounded cohomology; (b) to harvest the resulting applications in rigidity theory. Of central importance is the new interplay between measure theory, amenability, Banach representations on one hand, with the homological apparatus on the other hand. The applications obtained in this text include rigidity for actions on Teichmüller spaces and homeomorphisms of the circle. The main tools include Poisson boundaries for random walks, spectral sequences, Zimmer-amenability, cohomological induction.

Let G be an irreducible uniform lattice in a higher semi-simple rank Lie group or algebraic group. We prove that any G-action on the circle by C1 diffeomorphisms is finite. This is achieved by showing that natural map from bounded to usual second cohomology is injective. The latter holds also for non-trivial unitary coefficients, and implies more finiteness results for G; for instance the stable commutator length vanishes. We prove the same theorems for certain lattices in products of trees.

Note: In case of rank one factors, see this Erratum.

Functions of hyperbolic type encode representations on real or complex hyperbolic spaces, usually infinite-dimensional.

These notes set up the complex case. As applications, we prove the existence of a non-trivial deformation family of representations of SU(1,n) and of its infinite-dimensional kin Is(HC). We further classify all the self-representations of Is(HC) that satisfy a compatibility condition for the subgroup Is(HR). It turns out in particular that translation lengths and Cartan arguments determine each other for these representations.

In the real case, we revisit earlier results and propose some further constructions. 

Contrary to the finite-dimensional case, the Möbius group admits interesting self-representations when infinite-dimensional. We construct and classify all these self-representations. The proofs are obtained in the equivalent setting of isometries of Lobachevsky spaces and use kernels of hyperbolic type, in analogy to the classical concepts of kernels of positive and negative type.

A natural analogue of the Krein–Milman theorem is shown to fail for CAT(0) spaces.

Non-positively curved spaces admitting a cocompact isometric action of an amenable group are investigated. A classification is established under the assumption that there is no global fixed point at infinity under the full isometry group. The visual boundary is then a spherical building. When the ambient space is geodesically complete, it must be a product of flats, symmetric spaces, biregular trees and Bruhat–Tits buildings.

We provide moreover a sufficient condition for a spherical building arising as the visual boundary of a proper CAT(0) space to be Moufang, and deduce that an irreducible locally finite Euclidean building of dimension ≥ 2 is a Bruhat–Tits building if and only if its automorphism group acts cocompactly and chamber-transitively at infinity.

We give a complete characterization of the locally compact groups that are non-elementary Gromov-hyperbolic and amenable. They coincide with the class of mapping tori of discrete or continuous one-parameter groups of compacting automorphisms. We moreover give a description of all Gromov-hyperbolic locally compact groups with a cocompact amenable subgroup: modulo a compact normal subgroup, these turn out to be either rank one simple Lie groups, or automorphism groups of semi-regular trees acting doubly transitively on the set of ends. As an application, we show that the class of hyperbolic locally compact groups with a geometrically finite non-uniform lattice, is very restricted. 

We construct a continuous family of non-isometric minimal proper CAT(-1) spaces on which the isometry group Isom(Hn) of the hyperbolic n-space acts minimally and cocompactly by isometries. This provides the first examples of non-standard CAT(0) model spaces for simple Lie groups. We also classify all continuous non-elementary actions of Isom(Hn) on the infinite-dimensional real hyperbolic space.

Please see the addendum for additional information on Proposition 5.4. 

We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residual finiteness of lattices is also studied. Riemannian symmetric spaces are characterised amongst CAT(0) spaces admitting lattices in terms of the existence of parabolic isometries.

Note: Please see this erratum/addenda concerning Theorem 1.3.

We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and Bruhat–Tits buildings. Applications to discrete groups and further developments on non-positively curved lattices are exposed in a companion paper (above).

We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard manifolds are established for singular spaces; new arithmeticity and rigidity statements are obtained.

For every hyperbolic group and more general hyperbolic graphs, we construct an equivariant ideal bicombing: this is a homological analogue of the geodesic flow on negatively curved manifolds. We then construct a cohomological invariant which implies that several Measure Equivalence and Orbit Equivalence rigidity results established by Monod–Shalom hold for all non-elementary hyperbolic groups and their non-elementary subgroups. We also derive superrigidity results for actions of general irreducible lattices on a large class of hyperbolic metric spaces.

A note presenting a selection of results that are elaborated upon in Cocycle superrigidity and bounded cohomology for negatively curved spaces and Orbit equivalence rigidity and bounded cohomology. Proofs are given for illustrative “toy-cases”.

We show that a large family of groups without non-abelian free subgroups satisfy the following strengthening of non-amenability: they each have a rich supply of irreducible representations defining exotic C*-algebras. The construction is explicit.

This is an appendinx to Eli Glasner's article "On a question of Kazhdan and Yom Din". We show that the question of Kazhdan and Yom Din admits a negative answer for the space of (continuous) operators in Hilbert space. Namely, there exists operators that are nearly fixed by a group without being close to a fixed operator. Our construction is based on ideas introduced by Bożejko and Fendler in 1991.

Every Gelfand pair (G,K) admits a decomposition G=KP, where P<G is an amenable subgroup. In particular, the Furstenberg boundary of G is homogeneous.

Applications include the classification of non-positively curved spaces admitting Gelfand pairs, relying on earlier joint work with Caprace, as well as a canonical family of pure spherical functions in the sense of Gelfand–Godement for general Gelfand pairs.

We establish new results on the weak containment of quasi-regular and Koopman representations of a second countable locally compact group G associated with non-singular G-spaces. We deduce that any two boundary representations of a hyperbolic locally compact group are weakly equivalent. We also show that non-amenable hyperbolic locally compact groups with a cocompact amenable subgroup are characterized by the property that any two proper length functions are homothetic up to an additive constant. Combining those results with the work of Garncarek on the irreducibility of boundary representations of discrete hyperbolic groups, we deduce that a type I hyperbolic group with a cocompact lattice contains a cocompact amenable subgroup. Specializing to groups acting on trees, we answer a question of C. Houdayer and S. Raum.

Given a group Γ, we establish a connection between the unitarisability of its uniformly bounded representations and the asymptotic behaviour of the isoperimetric constants of Cayley graphs of Γ for increasingly large generating sets.
The connection hinges on an analytic invariant Lit(Γ) ∈ [0, ∞] which we call the Littlewood exponent. Finiteness, amenability, unitarisability and the existence of free subgroups are related respectively to the thresholds 0, 1, 2 and ∞ for Lit(Γ). Using graphical small cancellation theory, we prove that there exist groups Γ for which 1<Lit(Γ)<∞. Further applications, examples and problems are discussed.

For any group G containing an infinite elementary amenable subgroup, and any 2<p<∞, there exists closed invariant subspaces Ei ↑ ℓpG and F≠ 0 such that Ei∩ F = 0 for all i. This is an obstacle to ℓp-dimension and gives a negative answer to a question of Gaboriau.

J. Dixmier asked in 1950 whether every non-amenable group admits uniformly bounded representations that cannot be unitarised. We provide such representations upon passing to extensions by abelian groups. This gives a new characterisation of amenability. Furthermore, we deduce that certain Burnside groups are non-unitarisable, answering a question raised by G. Pisier.

We establish a connection between Dixmier's unitarisability problem and the expected degree of random forests on a group. As a consequence, a residually finite group is non-unitarisable if its first L2-Betti number is non-zero or if it is finitely generated with non-trivial cost. Our criterion also applies to torsion groups constructed by D. Osin, thus providing the first examples of non-unitarisable groups not containing a non-Abelian free subgroup.

Furstenberg has associated to every topological group G a universal boundary ∂(G). If we consider in addition a subgroup H<G, the relative notion of (G,H)-boundaries admits again a maximal object ∂(G,H). In the case of discrete groups, an equivalent notion was introduced by Bearden—Kalantar as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex Δ(G,H), namely the simplex of measures on ∂(G,H). We determine the boundary ∂(G,H) in a number of cases, highlighting properties that might appear unexpected.

We discuss equivariance for linear liftings of measurable functions. Existence is established when a transformation group acts amenably, as e.g. the Möbius group of the projective line.

Since the general proof is very simple but not explicit, we also provide a much more explicit lifting for semi-simple Lie groups acting on their Furstenberg boundary, using unrestricted Fatou convergence. This setting is relevant to L∞-cocycles for characteristic classes. 

Grigorchuk and Medynets recently conjectured that the topological full group of a minimal Cantor Z-action is amenable. They asked whether the statement holds for all minimal Cantor actions of general amenable groups as well. We answer in the negative by producing a minimal Cantor Z2-action for which the topological full group contains a non-abelian free group.

We analyse volume-preserving actions of product groups on Riemannian manifolds. Under a natural spectral irreducibility assumption, we prove the following dichotomy: Either the action is measurably isometric, in which case there are at most two factors; or the action is infinitesimally linear, which means that the derivative cocycle arises from unbounded linear representations of all factors.

As a first application, this provides lower bounds on the dimension of the manifold in terms of the number of factors in the acting group. Another application is a strong restriction for actions of non-linear groups. We prove our results by means of a new cocycle superrigidity theorem of independent interest, in analogy to Zimmer's programme. 

We establish new results and introduce new methods in the theory of measurable orbit equivalence. Our rigidity statements hold for a wide (uncountable) class of negatively curved groups. Amongst our applications are (a) measurable Mostow-type rigidity theorems for products of negatively curved groups; (b) prime factorization results for measure equivalence; (c) superrigidity for orbit equivalence; (d) the first examples of continua of type II1 equivalence relations with trivial outer automorphism group that are mutually not stably isomorphic.

This collection of expository articles by a range of established experts and newer researchers provides an overview of the recent developments in the theory of locally compact groups. It includes introductory articles on totally disconnected locally compact groups, profinite groups, p-adic Lie groups and the metric geometry of locally compact groups. Concrete examples, including groups acting on trees and Neretin groups, are discussed in detail. An outline of the emerging structure theory of locally compact groups beyond the connected case is presented through three complementary approaches: Willis' theory of the scale function, global decompositions by means of subnormal series, and the local approach relying on the structure lattice. An introduction to lattices, invariant random subgroups and L2-invariants, and a brief account of the Burger–Mozes construction of simple lattices are also included. A final chapter collects various problems suggesting future research directions.

This text is the preprint version of the concluding chapter for the book New Directions in Locally Compact Groups published by Cambridge University Press in the series Lecture Notes of the LMS. The recent progress on locally compact groups surveyed in that volume also reveals the considerable extent of the unexplored territories. Therefore, we wish to conclude it by mentioning a few open problems related to the material covered in the book and that we consider important at the time of this writing.

It is a well-known open problem since the 1970s whether a finitely generated perfect group can be normally generated by a single element or not. We prove that the topological version of this problem has an affirmative answer as long as we exclude infinite discrete quotients (which is probably a necessary restriction).

We present a contribution to the structure theory of locally compact groups. The emphasis is put on compactly generated locally compact groups which admit no infinite discrete quotient. It is shown that such a group possesses a characteristic cocompact subgroup which is either connected or admits a non-compact non-discrete topologically simple quotient. We also provide a complete description of groups all of whose proper quotients are compact, of characteristically simple groups and of groups admitting a subnormal series with all subquotients compact, or compactly generated Abelian, or compactly generated and topologically simple.

Two appendices introduce results and examples around the concept of quasi-product.

Update: please see correction below.

We correct an error in Proposition 2.6 and provide the amendments and additional arguments needed as a consequence of this change.

The automorphism group of a regular locally finite tree is shown to admit irreducible Banach representations that are not admissible. The dense subspace of smooth vectors contains no algebraically irreducible component. 

Given a transitive permutation group, a fundamental object for studying its higher transitivity properties is the permutation action of its isotropy subgroup. We reverse this relationship and introduce a universal construction of infinite permutation groups that takes as input a given system of imprimitivity for its isotropy subgroup.

This produces vast families of kaleidoscopic groups. We investigate their algebraic properties, such as simplicity and oligomorphy; their homological properties, such as acyclicity or contrariwise large Schur multipliers; their topological properties, such as unique polishability.

Our construction is carried out within the framework of homeomorphism groups of topological dendrites.

Let G be the homeomorphism group of a dendrite. We study the normal subgroups of G. For instance, there are uncountably many non-isomorphic such groups G that are simple groups. Moreover, these groups can be chosen so that any isometric G-action on any metric space has a bounded orbit. In particular they have the fixed point property (FH).

We establish obstructions for groups to act by homeomorphisms on dendrites. For instance, lattices in higher rank simple Lie groups will always fix a point or a pair. The same holds for irreducible lattices in products of connected groups. Further results include a Tits alternative and a description of the topological dynamics.

We briefly discuss to what extent our results hold for more general topological curves. 

We study isometric actions of tree automorphism groups on the infinite-dimensional hyperbolic spaces. On the one hand, we exhibit a general one-parameter family of such representations and analyse the corresponding equivariant embeddings of the trees, showing that they are convex-cocompact and asymptotically isometric. On the other hand, focusing on the case of sufficiently transitive groups of automorphisms of locally finite trees, we classify completely all irreducible representations by isometries of hyperbolic spaces. It turns out that in this case our one-parameter family exhausts all non-elementary representations.

Please see the addendum for a small insecticide for Lemma 6.5. 

Let Γ be an irreducible lattice in a product of n infinite irreducible complete Kac–Moody groups of simply laced type over finite fields. We show that if n≥3, then each Kac–Moody groups is in fact a simple algebraic group over a local field and Γ is an arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n≥2: either Γ is an S-arithmetic (hence linear) group, or Γ is not residually finite. In that case, it is even virtually simple when the ground field is large enough.

More general CAT(0) groups are also considered throughout. 

We prove general superrigidity results for actions of irreducible lattices on CAT(0) spaces; first, in terms of the ideal boundary, and then for the intrinsic geometry (including for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform irreducible lattices in non-simple groups. The proofs rely on simple geometric arguments, including a splitting theorem which can be viewed as an infinite-dimensional (and singular) generalization of the Lawson–Yau/Gromoll–Wolf theorem. Appendix A gives a very elementary proof of commensurator superrigidity; Appendix B proves that all our results also hold for certain non-uniform lattices. 

Note: Example 18 is corrected here. 

We establish an arithmeticity vs. non-linearity alternative for irreducible lattices in suitable product groups, such as for instance products of topologically simple groups. This applies notably to (a large class of) Kac–Moody groups. The alternative relies on a CAT(0) superrigidity theorem, as we follow Margulis' reduction of arithmeticity to superrigidity.

A note announcing my results published in JAMS 2006.

We introduce new techniques to extend superrigidity theory beyond the scope of Lie or algebraic groups. We construct a cohomological invariant which accounts for, and generalizes, all known superrigidity results for actions on negatively curved spaces. Together with a new vanishing result and the machinery of bounded cohomology, this enables us to prove a general superrigidity theorem for actions of irreducible lattices on spaces of negative curvature. We also prove a cocycle version à la Zimmer.

Ulam asked whether all Lie groups can be represented faithfully on a countable set. We establish a reduction of Ulam's problem to the case of simple Lie groups.

In particular, we solve the problem for all solvable Lie groups and more generally Lie groups with a linear Levi component. It follows that every amenable locally compact second countable group acts faithfully on a countable set. 

We investigate the group G∨H obtained by gluing together two groups G and H at the neutral element. This construction curiously shares some properties with the free product but others with the direct product. Our results address among others Property (T), CAT(0) cubical complexes, local embeddability, amenable actions, and the algebraic structure of G∨H.

Ulam asked whether every connected Lie group can be represented on a countable structure. This is known in the linear case. We establish it for the first family of non-linear groups, namely in the nilpotent case. Further context is discussed to illustrate the relevance of nilpotent groups for Ulam's problem.

We propose elementary and explicit presentations of groups that have no amenable quotients and yet are SQ-universal. Examples include groups with a finite classifying space, no Kazhdan subgroups and no Haagerup quotients.

Recall that a rng is a ring which is possibly non-unital. In this note, we address the problem whether every finitely generated idempotent rng (abbreviated as irng) is singly generated as an ideal. It is well-known that it is the case for a commutative irng. We prove here it is also the case for a free rng on finitely many idempotents and for a finite irng. A relation to the Wiegold problem for perfect groups is discussed.

A note presenting the first examples of groups that are boundedly generated, have property (T) and have a one-dimensional space of pseudocharacters.