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.
Highlights
The sphere at infinity ∂∞ X of a negatively curved symmetric space X carries many rich structures, from the geometric, analytic and arithmetic points of view
Arithmetic subgroups of the isometry group of X endow the sphere at infinity of X with arithmetic structures, and problems of equidistribution of rational points or subvarieties in ∂∞ X, as well as in other homogeneous manifolds, have been intensively studied
We study the quaternionic hyperbolic spaces X, whose extreme rigidity is exemplified by the Margulis–Gromov–Schoen theorem in [13], proving, contrarily to the real or complex case, the arithmeticity of lattices in the isometry group of X
Summary
The sphere at infinity ∂∞ X of a negatively curved symmetric space X carries many rich structures, from the geometric, analytic and arithmetic points of view. We study the quaternionic hyperbolic spaces X, whose extreme rigidity is exemplified by the Margulis–Gromov–Schoen theorem in [13], proving, contrarily to the real or complex case, the arithmeticity of lattices in the isometry group of X. Paulin chains in orbits of arithmetic groups built using maximal orders in rational quaternion algebras. Let q be the quaternionic Hermitian form on the right vector space H3 over H defined by q(z0, z1, z2) = − tr(z0z2) + n(z1), and PUq its projective unitary group. The following result proves that the centers of the arithmetic chains in an orbit under the arithmetic group PUq (O) equidistribute in the quaternionic Heisenberg group.
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