Ambrosio and Kirchheim presented a theory of currents with finite mass in complete metric spaces. We develop a variant of the theory that does not rely on a finite mass condition, closely paralleling the classical Federer-Fleming theory. If the underlying metric space is an open subset of a Euclidean space, we obtain a natural chain monomorphism from general metric currents to general classical currents whose image contains the locally flat chains and which restricts to an isomorphism for locally normal currents. We give a detailed exposition of the slicing theory for locally normal currents with respect to locally Lipschitz maps, including the rectifiable slices theorem, and of the compactness theorem for locally integral currents in locally compact metric spaces, assuming only standard results from analysis and measure theory.

Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the '60s Isbell showed that every metric space X has an injective hull E(X). Here it is proved that if X is the vertex set of a connected locally finite graph with a uniform stability property of intervals, then E(X) is a locally finite polyhedral complex with finitely many isometry types of n-cells, isometric to polytopes in l n ∞ , for each n. This applies to a class of finitely generated groups Γ, including all word hyperbolic groups and abelian groups, among others. Then Γ acts properly on E(Γ) by cellular isometries, and the first barycentric subdivision of E(Γ) is a model for the classifying space EΓ for proper actions. If Γ is hyperbolic, E(Γ) is finite dimensional and the action is cocompact. In particular, every hyperbolic group acts properly and cocompactly on a space of non-positive curvature in a weak (but non-coarse) sense.finite then the injective hull is a finite polyhedral complex of dimension at most 1 2 |X| whose n-cells are isometric to polytopes in l n ∞ = l ∞ ({1, . . . , n}). A detailed account of injective metric spaces and hulls is given below, in Secs. 2 and 3.Isbell's construction was rediscovered 20 years later by Dress [13] (and even another time in [10]). Due to this independent work and a characterization of injective metric spaces from [3], metric injective hulls are also called tight spans or hyperconvex hulls in the literature, furthermore "hull" is often substituted by "envelope". Tight spans are widely known in discrete mathematics and have notably been used in phylogenetic analysis (see [17,16] for some surveys). Apart from the two foundational papers [24,13] and some work referring to Banach spaces (see, for instance, [25,36,11]), the vast literature on metric injective hulls deals almost exclusively with finite metric spaces. Dress proved that for certain discrete metric spaces X the tight span T X still has a polyhedral structure (see Items (5.19), (6.2), and (6.6) in [13]); these results, however, presuppose that T X is locally finite dimensional. A simple sufficient, geometric condition on X to this effect has been missing (but see Theorem 9 and Item (5.14) in [13]).Here it is now shown that, in the case of integer valued metrics, a weak form of the fellow traveler property for discrete geodesics serves the purpose and even ensures that E(X) is proper, provided X is; see Theorem 1.1 below. The polyhedral structure of E(X) and the possible isometry types of cells are described in detail and no prior knowledge of the constructions in [24, 13] is assumed. With regard to applications in geometric group theory, a general fixed point theorem for injective metric spaces is pointed out (Proposition 1.2), which closely parallels the wellknown result for CAT(0) spaces. It has been known for some time that if the metric space X is δ-hyperbolic, then so is E(X), and this implies that E(X) is within finite distance of e(X), provided X is geodesic or discretely geodesic (Propos...

We generalize Kirszbraun's extension theorem for Lipschitz maps between (subsets of) euclidean spaces to metric spaces with upper or lower curvature bounds in the sense of A.D. Alexandrov. As a byproduct we develop new tools in the theory of tangent cones of these spaces and obtain new characterization results which may be of independent interest.

A geodesic bicombing on a metric space selects for every pair of points a geodesic connecting them. We prove existence and uniqueness results for geodesic bicombings satisfying different convexity conditions. In combination with recent work by the second author on injective hulls, this shows that every word hyperbolic group acts geometrically on a proper, finite dimensional space X with a unique (hence equivariant) convex geodesic bicombing of the strongest type. Furthermore, the Gromov boundary of X is a Z-set in the closure of X, and the latter is a metrizable absolute retract, in analogy with the Bestvina-Mess theorem on the Rips complex.

Recently, a new embedding/compactness theorem for integral currents in a sequence of metric spaces has been established by the second author. We present a version of this result for locally integral currents in a sequence of pointed metric spaces. To this end we introduce another variant of the Ambrosio-Kirchheim theory of currents in metric spaces, including currents with finite mass in bounded sets.