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Urs Lang

Publications and source records attributed to Urs Lang.

14 recordsLinked to original sources
arXiv2026

Isoperimetric inequalities in Hadamard spaces of asymptotic rank two

Gromov's isoperimetric gap conjecture for Hadamard spaces states that cycles in dimensions greater than or equal to the asymptotic rank admit linear isoperimetric filling inequalities, as opposed to the inequalities of Euclidean type in lower dimensions. In the case of asymptotic rank 2, recent progress was made by Druţu-Lang-Papasoglu-Stadler who established a homotopical inequality for Lipschitz 2-spheres with exponents arbitrarily close to 1. We prove a homological inequality of the same type for general cycles in dimensions at least 2, assuming that the ambient space has finite linearly controlled asymptotic dimension. This holds in particular for all Hadamard 3-manifolds and finite-dimensional CAT(0) cube complexes.

math.MG↗
arXiv2025

Minimal tetrahedra and an isoperimetric gap theorem in non-positive curvature

We investigate isoperimetric inequalities for Lipschitz 2-spheres in CAT(0) spaces, proving bounds on the volume of efficient null-homotopies. In one dimension lower, it is known that a quadratic inequality with a constant smaller than $c_2=1/(4π)$ -- the optimal constant for the Euclidean plane -- implies that the underlying space is Gromov hyperbolic, and a linear inequality holds. We establish the first analogous gap theorem in higher dimensions: if a proper CAT(0) space satisfies a Euclidean inequality for 2-spheres with a constant below the sharp threshold $c_3=1/(6\sqrtπ)$, then the space also admits an inequality with an exponent arbitrarily close to 1. As a corollary we obtain a similar result for Lipschitz surfaces of higher genus. Towards our main theorem we prove a (non-sharp) Euclidean isoperimetric inequality for null-homotopies of 2-spheres, apparently missing in the literature. A novelty in our approach is the introduction of minimal tetrahedra, which we demonstrate satisfy a linear inequality.

math.MG↗
arXiv2023

A combinatorial higher-rank hyperbolicity condition

We investigate a coarse version of a $2(n+1)$-point inequality characterizing metric spaces of combinatorial dimension at most $n$ due to Dress. This condition, experimentally called $(n,δ)$-hyperbolicity, reduces to Gromov's quadruple definition of $δ$-hyperbolicity in case $n = 1$. The $l_\infty$-product of $n$ $δ$-hyperbolic spaces is $(n,δ)$-hyperbolic. Every $(n,δ)$-hyperbolic metric space, without any further assumptions, possesses a slim $(n+1)$-simplex property analogous to the slimness of quasi-geodesic triangles in Gromov hyperbolic spaces. In connection with recent work in geometric group theory, we show that every Helly group and every hierarchically hyperbolic group of (asymptotic) rank $n$ acts geometrically on some $(n,δ)$-hyperbolic space.

math.MG↗
arXiv2023

Characterizations of higher rank hyperbolicity

The concept of Gromov hyperbolicity manifests itself in many different ways. With only mild assumptions on the underlying metric space, the spectrum of equivalent properties includes various thin triangle conditions, the stability of quasi-geodesics (the Morse lemma), a linear isoperimetric filling inequality for closed curves, and a sub-quadratic isoperimetric inequality. We present a similar list of six equivalent properties in the context of generalized non-positive curvature and higher asymptotic rank. This complements results of Wenger and of Kleiner and the second author. We give a largely self-contained proof, providing some improvements and simplifications for the known part.

math.MG↗
arXiv2020

Geodesic spaces of low Nagata dimension

We show that every geodesic metric space admitting an injective continuous map into the plane as well as every planar graph has Nagata dimension at most two, hence asymptotic dimension at most two. This relies on and answers a question in a very recent work by Fujiwara and Papasoglu. We conclude that all three-dimensional Hadamard manifolds have Nagata dimension three. As a consequence, all such manifolds are absolute Lipschitz retracts.

math.MG↗
arXiv2019

Higher rank hyperbolicity

The large-scale geometry of hyperbolic metric spaces exhibits many distinctive features, such as the stability of quasi-geodesics (the Morse Lemma), the visibility property, and the homeomorphism between visual boundaries induced by a quasi-isometry. We prove a number of closely analogous results for spaces of rank $n \ge 2$ in an asymptotic sense, under some weak assumptions reminiscent of nonpositive curvature. For this purpose we replace quasi-geodesic lines with quasi-minimizing (locally finite) $n$-cycles of $r^n$ volume growth; prime examples include $n$-cycles associated with $n$-quasiflats. Solving an asymptotic Plateau problem and producing unique tangent cones at infinity for such cycles, we show in particular that every quasi-isometry between two proper CAT(0) spaces of asymptotic rank $n$ extends to a class of $(n-1)$-cycles in the Tits boundaries.

math.MG↗
arXiv2016

Flats in spaces with convex geodesic bicombings

In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales. We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function along distinguished geodesics. The discussion includes, among other things, the Flat Torus Theorem and Gromov's hyperbolicity criterion referring to embedded planes. This generalizes results of Bowditch for Busemann spaces.

math.MG↗
arXiv2014

Convex geodesic bicombings and hyperbolicity

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.

math.MG↗
arXiv2013

Metric stability of trees and tight spans

In this note, we prove optimal extension results for roughly isometric relations between metric (R-)trees and injective metric spaces. This yields sharp stability estimates, in terms of the Gromov-Hausdorff (GH) distance, for certain metric spanning constructions: The GH distance of two metric trees spanned by some subsets is smaller than or equal to the GH distance of these sets. The GH distance of the injective hulls, or tight spans, of two metric spaces is at most twice the GH distance between themselves.

math.MG↗
arXiv2012

On Toponogov's comparison theorem for Alexandrov spaces

In this expository note, we present a transparent proof of Toponogov's theorem for Alexandrov spaces in the general case, not assuming local compactness of the underlying metric space. More precisely, we show that if M is a complete geodesic metric space such that the Alexandrov triangle comparisons for curvature greater than or equal to k are satisfied locally, then these comparisons also hold in the large. The proof is a modification of an argument due to Plaut.

math.MG↗
arXiv2012

Injective hulls of certain discrete metric spaces and groups

Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the 1960es, J. R. 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_\infty, for each n. This applies to a class of finitely generated groups G, including all word hyperbolic groups and abelian groups, among others. Then G acts properly on E(G) by cellular isometries, and the first barycentric subdivision of E(G) is a model for the classifying space \underbar{E}G for proper actions. If G is hyperbolic, E(G) 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.

math.GR↗
arXiv2010

The pointed flat compactness theorem for locally integral currents

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.

math.DG↗
arXiv2004

Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions

We discuss a variation of Gromov's notion of asymptotic dimension that was introduced and named Nagata dimension by Assouad. The Nagata dimension turns out to be a quasisymmetry invariant of metric spaces. The class of metric spaces with finite Nagata dimension includes in particular all doubling spaces, metric trees, euclidean buildings, and homogeneous or pinched negatively curved Hadamard manifolds. Among others, we prove a quasisymmetric embedding theorem for spaces with finite Nagata dimension in the spirit of theorems of Assouad and Dranishnikov, and we characterize absolute Lipschitz retracts of finite Nagata dimension.

math.MG↗