Pierre-Emmanuel Caprace

Simplicity and superrigidity of twin building lattices

Kac–Moody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non-positively curved twinned buildings: those are the most important (but not the only) examples of twin building lattices. We prove that these lattices are simple if the corresponding buildings are irreducible and not of affine type (i.e. they are not Bruhat–Tits buildings). Many of them are finitely presented and enjoy property (T). Our arguments explain geometrically why simplicity fails to hold only for affine Kac–Moody groups. Moreover we prove that a nontrivial continuous homomorphism from a completed Kac–Moody group is always proper. We also show that Kac–Moody lattices fulfill conditions implying strong superrigidity properties for isometric actions on non-positively curved metric spaces. Most results apply to the general class of twin building lattices.

Rank Rigidity for Cat(0) Cube Complexes

We prove that any group acting essentially without a fixed point at infinity on an irreducible finite-dimensional CAT(0) cube complex contains a rankone isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube complexes. We derive a number of other consequences for CAT(0) cube complexes, including a purely geometric proof of the Tits alternative, an existence result for regular elements in (possibly non-uniform) lattices acting on cube complexes, and a characterization of products of trees in terms of bounded cohomology.

Decomposing locally compact groups into simple pieces

We present a contribution to the structure theory of locally compact groups. The emphasis is 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 description of characteristically simple groups and of groups all of whose proper quotients are compact. We show that Noetherian locally compact groups without infinite discrete quotient admit a subnormal series with all subquotients compact, compactly generated Abelian, or compactly generated topologically simple.

Isometry groups of non-positively curved spaces: structure theory

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 characterizing properties of symmetric spaces and Bruhat-Tits buildings. Applications to discrete groups and further developments on non-positively curved lattices are discussed in a companion paper [27].

Isometry groups of non-positively curved spaces: discrete subgroups

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 super-rigidity and arithmeticity of abstract lattices. Residual finiteness of lattices is also studied. Riemannian symmetric spaces are characterized amongst CAT(0) spaces admitting lattices in terms of the existence of parabolic isometries.

On 2-spherical Kac–Moody groups and their central extensions

Abstract We provide a new presentation for simply connected Kac-Moody groups of 2-spherical type and for their universal central extensions. Under mild local restrictions, these results extend to the more general class of groups of Kac-Moody type (i.e. groups endowed with a root datum).

Gelfand pairs and strong transitivity for Euclidean buildings

Let G be a locally compact group acting properly by type-preserving automorphisms on a locally finite thick Euclidean building X and K be the stabilizer of a special vertex in X. It is known that (G;K) is a Gelfand pair as soon as G acts strongly transitively on X; this is in particular the case when G is a semi-simple algebraic group over a local field. We show a converse of this statement, namely: if (G;K) is a Gelfand pair and G acts cocompactly on X, then the action is strongly transitive. The proof uses the existence of strongly regular hyperbolic elements in G and their peculiar dynamics on the spherical building at innity. Other equivalent formulations are also obtained, including the fact that G is strongly transitive on X if and only if it is strongly transitive on the spherical building at infinity.

At infinity of finite-dimensional CAT(0) spaces

We show that any filtering family of closed convex subsets of a finite-dimensional CAT(0) space X has a non-empty intersection in the visual bordification

Locally normal subgroups of totally disconnected groups. Part II: Compactly generated simple groups

{ \overline{X} = X \cup \partial X}