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Geometry of compactifications of locally symmetric spaces

Lizhen Ji, Robert Macpherson (2002)

Annales de l’institut Fourier

For a locally symmetric space M , we define a compactification M ∪ M ( ∞ ) which we call the “geodesic compactification”. It is constructed by adding limit points in M ( ∞ ) to certain geodesics in M . The geodesic compactification arises in other contexts. Two general constructions of Gromov for an ideal boundary of a Riemannian manifold give M ( ∞ ) for locally symmetric spaces. Moreover, M ( ∞ ) has a natural group theoretic construction using the Tits building. The geodesic compactification plays two fundamental roles in...

Hilbert volume in metric spaces. Part 1

Misha Gromov (2012)

Open Mathematics

We introduce a notion of Hilbertian n-volume in metric spaces with Besicovitch-type inequalities built-in into the definitions. The present Part 1 of the article is, for the most part, dedicated to the reformulation of known results in our terms with proofs being reduced to (almost) pure tautologies. If there is any novelty in the paper, this is in forging certain terminology which, ultimately, may turn useful in an Alexandrov kind of approach to singular spaces with positive scalar curvature [Gromov...

Isometric Embeddings of Pro-Euclidean Spaces

Barry Minemyer (2015)

Analysis and Geometry in Metric Spaces

In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank...

Isometries of systolic spaces

Tomasz Elsner (2009)

Fundamenta Mathematicae

We provide a classification of isometries of systolic complexes corresponding to the classification of isometries of CAT(0)-spaces. We prove that any isometry of a systolic complex either fixes the barycentre of some simplex (elliptic case) or stabilizes a thick geodesic (hyperbolic case). This leads to an alternative proof of the fact that finitely generated abelian subgroups of systolic groups are undistorted.

Large-scale isoperimetry on locally compact groups and applications

Romain Tessera (2006/2007)

Séminaire de théorie spectrale et géométrie

We introduce various notions of large-scale isoperimetric profile on a locally compact, compactly generated amenable group. These asymptotic quantities provide measurements of the degree of amenability of the group. We are particularly interested in a class of groups with exponential volume growth which are the most amenable possible in that sense. We show that these groups share various interesting properties such as the speed of on-diagonal decay of random walks, the vanishing of the reduced first...

Les géométries de Hilbert sont à géométrie locale bornée

Bruno Colbois, Constantin Vernicos (2007)

Annales de l’institut Fourier

On montre que la géométrie de Hilbert d’un domaine convexe de ℝ n est à géométrie locale bornée c-à-d que pour un rayon fixé, toutes les boules sont bilipschitz à un domaine de ℝ n euclidien. On en déduit que si la géométrie de Hilbert est hyperbolique au sens de Gromov, alors le bas de son spectre est strictement positif. On donne un contre-exemple en dimension trois qui montre que la réciproque n’est pas vraie pour les géométries de Hilbert non planes.

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