Rigidity, counting and equidistribution of quaternionic Cartan chains

TL;DR

The paper proves a Cartan-type rigidity for chain-preserving maps on the boundary of quaternionic hyperbolic space, showing that such maps come from projective unitary transformations. It then develops a counting and equidistribution theory for arithmetic chains in the quaternionic Heisenberg group, with explicit asymptotics governed by quaternionic arithmetic data. By leveraging the hyperbolic geometry, Siegel domain model, and the hyper CR structure, it provides both a rigidity result and quantitative arithmetic results, including a precise $\epsilon^{-10}$ law for the growth of chain counts and Haar-measure equidistribution of chain centers. These results advance understanding of rigidity and distribution phenomena in quaternionic hyperbolic geometry and its arithmetic lattices, with potential applications to hyperspherical geometry and automorphic forms in this setting.

Abstract

We prove an analog of Cartan's theorem, saying that the chain-preserving transformations of the boundary of the quaternionic hyperbolic spaces are projective transformations. We give a counting and equidistribution result for the orbits of arithmetic chains in the quaternionic Heisenberg group.

Theorems & Definitions (12)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Proposition 3.1
• Lemma 3.2
• Theorem 3.3
• Proposition 3.4
• Theorem 4.1
• Theorem 4.2