## Journal of Differential Geometry

### Symmetry gaps in Riemannian geometry and minimal orbifolds

Wouter van Limbeek

#### Abstract

We study the size of the isometry group $\mathrm{Isom}(M,g)$ of Riemannian manifolds $(M,g)$ as $g$ varies. For $M$ not admitting a circle action, we show that the order of $\mathrm{Isom}(M,g)$ can be universally bounded in terms of the bounds on Ricci curvature, diameter, and injectivity radius of $M$. This generalizes results known for negative Ricci curvature to all manifolds.

More generally we establish a similar universal bound on the index of the deck group $\pi_1 (M)$ in the isometry group $\mathrm{Isom}(\widetilde{M},\widetilde{g})$ of the universal cover $\widetilde{M}$ in the absence of suitable actions by connected groups. We apply this to characterize locally symmetric spaces by their symmetry in covers. This proves a conjecture of Farb andWeinberger with the additional assumption of bounds on curvature, diameter, and injectivity radius. Further we generalize results of KazhdanñMargulis and Gromov on minimal orbifolds of nonpositively curved manifolds to arbitrary manifolds with only a purely topological assumption.

#### Article information

Source
J. Differential Geom., Volume 105, Number 3 (2017), 487-517.

Dates
Received: 19 May 2014
First available in Project Euclid: 3 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1488503005

Digital Object Identifier
doi:10.4310/jdg/1488503005

Mathematical Reviews number (MathSciNet)
MR3619310

Zentralblatt MATH identifier
1364.53042

#### Citation

van Limbeek, Wouter. Symmetry gaps in Riemannian geometry and minimal orbifolds. J. Differential Geom. 105 (2017), no. 3, 487--517. doi:10.4310/jdg/1488503005. https://projecteuclid.org/euclid.jdg/1488503005