On the volume of orbifold quotients of symmetric spaces

TL;DR

This paper derives explicit lower volume bounds for orbifold quotients of symmetric spaces using curvature estimates and volume comparison, extending known results to higher dimensions and specific cases like the octonionic hyperbolic plane.

Contribution

It provides the first explicit volume bound for the octonionic hyperbolic plane and improves bounds for higher-dimensional hyperbolic orbifolds.

Findings

First explicit volume bound for octonionic hyperbolic orbifolds

Improved lower bounds for hyperbolic orbifolds in dimensions > 3

Unified approach using curvature bounds and volume comparison

Abstract

A classic theorem of Kazhdan and Margulis states that for any semisimple Lie group without compact factors, there is a positive lower bound on the covolume of lattices. H. C. Wang's subsequent quantitative analysis showed that the fundamental domain of any lattice contains a ball whose radius depends only on the group itself. A direct consequence is a positive minimum volume for orbifolds modeled on the corresponding symmetric space. However, sharp bounds are known only for hyperbolic orbifolds of dimensions two and three, and recently for quaternionic hyperbolic orbifolds of all dimensions. As in arXiv:0911.4712 and arXiv:1205.2011, this article combines H. C. Wang's radius estimate with an improved upper sectional curvature bound for a canonical left-invariant metric on a real semisimple Lie group and uses Gunther's volume comparison theorem to deduce an explicit uniform lower volume bound for arbitrary orbifold quotients of a given irreducible symmetric spaces of non-compact type. The numerical bound for the octonionic hyperbolic plane is the first such bound to be given. For (real) hyperbolic orbifolds of dimension greater than three, the bounds are an improvement over what was previously known.