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.

Abstract. 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 cusp-uniform non-uniform lattice, is very restricted.

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 and only if the corresponding buildings are (irreducible and) not of affine type (i.e. they are not Bruhat-Tits buildings). In fact, 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

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

We use the structure lattice, introduced in Part I, to undertake a systematic study of the class S consisting of compactly generated, topologically simple, totally disconnected locally compact groups that are nondiscrete. Given G ∈ S , we show that compact open subgroups of G involve finitely many isomorphism types of composition factors, and do not have any soluble normal subgroup other than the trivial one. By results of Part I, this implies that the centralizer lattice and local decomposition lattice of G are Boolean algebras. We show that the G-action on the Stone space of those Boolean algebras is minimal, strongly proximal, and microsupported. Building upon those results, we obtain partial answers to the following key problems: Are all groups in S abstractly simple? Can a group in S be amenable? Can a group in S be such that the contraction groups of all of its elements are trivial? 2010 Mathematics Subject Classification: 22D05 (primary); 20E15, 20E32 (secondary)