A diameter gap for quotients of the unit sphere

Gorodski, Claudio; Lange, Christian; Lytchak, Alexander; Mendes, Ricardo A. E.

doi:10.4171/jems/1272

Cited by 4 publications

(1 citation statement)

References 20 publications

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“…Then the quotient space M/G is a metric space whose metric properties are often related to properties of the action in an interesting way, see e.g. [GLLM22,LT10]. Usually, this quotient is not a Riemannian manifold, but an Alexandrov space with curvature locally bounded from below.…”

Section: Introductionmentioning

confidence: 99%

Orbifolds and manifold quotients with upper curvature bounds

Lange¹

2023

Preprint

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We characterize Riemannian orbifolds with an upper curvature bound in the Alexandrov sense as reflectofolds, i.e. Riemannian orbifolds all of whose local groups are generated by reflections, with the same upper bound on the sectional curvature. Combined with a result by Lytchak-Thorbergsson this implies that a quotient of a Riemannian manifold by a closed group of isometries has locally bounded curvature (from above) in the Alexandrov sense if and only if it is a reflectofold.

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Section: Introductionmentioning

confidence: 99%

Orbifolds and manifold quotients with upper curvature bounds

Lange¹

2023

Preprint

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Orbifolds and Manifold Quotients with Upper Curvature Bounds

Lange

2024

Transformation Groups

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We characterize Riemannian orbifolds with an upper curvature bound in the Alexandrov sense as reflectofolds, i.e., Riemannian orbifolds all of whose local groups are generated by reflections, with the same upper bound on the sectional curvature. Combined with a result by Lytchak–Thorbergsson this implies that a quotient of a Riemannian manifold by a closed group of isometries has locally bounded curvature (from above) in the Alexandrov sense if and only if it is a reflectofold.

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Fundamental groups and group presentations with bounded relator lengths

Zamora

2024

Geom Dedicata

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We study the geometry of compact geodesic spaces with trivial first Betti number admitting large finite groups of isometries. We show that if a finite group G acts by isometries on a compact geodesic space X whose first Betti number vanishes, then $${\text {diam}}(X) / {\text {diam}}(X / G ) \le 4 \sqrt{ \vert G \vert }$$ diam ( X ) / diam ( X / G ) ≤ 4 | G | . For a group G and a finite symmetric generating set S, $$P_k(\varGamma (G, S))$$ P k ( Γ ( G , S ) ) denotes the 2-dimensional CW-complex whose 1-skeleton is the Cayley graph $$\varGamma $$ Γ of G with respect to S and whose 2-cells are m-gons for $$0 \le m \le k$$ 0 ≤ m ≤ k , defined by the simple graph loops of length m in $$\varGamma $$ Γ , up to cyclic permutations. Let G be a finite abelian group with $$\vert G \vert \ge 3$$ | G | ≥ 3 and S a symmetric set of generators for which $$P_k(\varGamma (G,S))$$ P k ( Γ ( G , S ) ) has trivial first Betti number. We show that the first nontrivial eigenvalue $$-\lambda _1$$ - λ 1 of the Laplacian on the Cayley graph satisfies $$\lambda _1 \ge 2 - 2 \cos ( 2 \pi / k ) $$ λ 1 ≥ 2 - 2 cos ( 2 π / k ) . We also give an explicit upper bound on the diameter of the Cayley graph of G with respect to S of the form $$O (k^2 \vert S \vert \log \vert G \vert )$$ O ( k 2 | S | log | G | ) . Related explicit bounds for the Cheeger constant and Kazhdan constant of the pair (G, S) are also obtained.

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A diameter gap for quotients of the unit sphere

Cited by 4 publications

References 20 publications

Orbifolds and manifold quotients with upper curvature bounds

Orbifolds and manifold quotients with upper curvature bounds

Orbifolds and Manifold Quotients with Upper Curvature Bounds

Fundamental groups and group presentations with bounded relator lengths

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