Quaternionic Hyperbolic Space Research Articles

Characterising trees and hyperbolic spaces by their boundaries

We use the language of proper CAT(-1) spaces to study thick, locally compact trees, the real, complex and quaternionic hyperbolic spaces and the hyperbolic plane over the octonions. These are rank 1 Euclidean buildings, respectively rank 1 symmetric spaces of non-compact type. We give a uniform proof that these spaces may be reconstructed using the cross ratio on their visual boundary, bringing together the work of Tits and Bourdon.

A characterization of the L2-range of the Poisson transforms on a class of vector bundles over the quaternionic hyperbolic spaces

We study the L2-boundedness of the Poisson transforms associated to the homogeneous vector bundles Sp(n,1)×Sp(n)×Sp(1)Vτ over the quaternionic hyperbolic spaces Sp(n,1)/Sp(n)×Sp(1) associated with irreducible representations τ of Sp(n)×Sp(1) which are trivial on Sp(n). As a consequence, we describe the image of the section space L2(Sp(n,1)×Sp(n)×Sp(1)Vτ) under the generalized spectral projections associated to a family of eigensections of the Casimir operator.

Sharp Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane

The main purpose of this paper is to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane using the Helgason–Fourier analysis on symmetric spaces. A crucial part of our work is to establish appropriate factorization theorems on these spaces, which can be of independent interest. To this end, we need to identify and introduce the “quaternionic Geller operators” and the “octonionic Geller operators”, which have been absent on these spaces. Combining the factorization theorems and the Geller type operators with the Helgason–Fourier analysis on symmetric spaces, some precise estimates for the heat and the Bessel–Green– Riesz kernels, and the Kunze–Stein phenomenon for connected real simple groups of real rank one with finite center, we succeed to establish the higher order Poincaré– Sobolev and Hardy–Sobolev–Maz’ya inequalities on quaternionic hyperbolic spaces and on the Cayley hyperbolic plane. The kernel estimates required to prove these inequalities are also sufficient to establish the Adams and Hardy–Adams inequalities on these spaces. This paper, together with our earlier works on real and complex hyperbolic spaces, completes our study of the factorization theorems, higher order Poincaré–Sobolev, Hardy–Sobolev–Maz’ya, Adams and Hardy–Adams inequalities on all rank one symmetric spaces of noncompact type.

Nonhomogeneous expanding flows in hyperbolic spaces

In the present paper, we consider star-shaped mean convex hypersurfaces of the real, complex and quaternionic hyperbolic space evolving by a class of nonhomogeneous expanding flows. For any choice of the ambient manifold, the initial conditions are preserved and the long-time existence of the flow is proved. The geometry of the ambient space influences the asymptotic behaviour of the flow: after a suitable rescaling, the induced metric converges to a conformal multiple of the standard Riemannian round metric of the sphere if the ambient manifold is the real hyperbolic space; otherwise, it converges to a conformal multiple of the standard sub-Riemannian metric on the odd-dimensional sphere. Finally, in every case, we are able to construct infinitely many examples such that the limit does not have constant scalar curvature.

INTEGRABILITY OF QUATERNION-KÄHLER SYMMETRIC SPACES

We find a necessary condition for the existence of an action of a Lie group G by quaternionic automorphisms on an integrable quaternionic manifold in terms of representations of 𝔤. We check this condition and prove that a Riemannian symmetric space of dimension 4n for n ≥ 2 has an invariant integrable almost quaternionic structure if and only if it is quaternionic vector space, quaternionic hyperbolic space, or quaternionic projective space.

Rigidity, counting and equidistribution of quaternionic Cartan chains

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.

Homogeneous and inhomogeneous isoparametric hypersurfaces in rank one symmetric spaces

Abstract We conclude the classification of cohomogeneity one actions on symmetric spaces of rank one by classifying cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence. As a by-product of our proof, we produce uncountably many examples of inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures in quaternionic hyperbolic spaces.

Counting and equidistribution in quaternionic Heisenberg groups

AbstractWe develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebraAover${\mathbb{Q}}$in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points overAin quaternionic Heisenberg groups.

Local rigidity of certain actions of solvable groups on the boundaries of rank-one symmetric spaces

Let $ G $ be the group of orientation-preserving isometries of a rank-one symmetric space $ X $ of non-compact type. We study local rigidity of certain actions of a solvable subgroup $ \Gamma \subset G $ on the boundary of $ X $, which is diffeomorphic to a sphere. When $ X $ is a quaternionic hyperbolic space or the Cayley hyperplane, the action we constructed is locally rigid.

The Moduli Space of Points in the Boundary of Quaternionic Hyperbolic Space

Let $\mathcal{F}_1(n,m)$ be the ${\rm PSp}(n,1)$-configuration space of ordered $m$-tuple of pairwise distinct points in the boundary of quaternionic hyperbolic n-space $\partial\mathbf{H}_\mathbb{H}^n$ , i.e., the $m$-tuple of pairwise distinct points in $\partial\mathbf{H}_\mathbb{H}^n$ up to the diagonal action of ${\rm PSp}(n,1)$. In terms of Cartan's angular invariant and cross-ratio invariants, the moduli space of $\mathcal{F}_1(n,m)$ is described by using Moore's determinant. We show that the moduli space of $\mathcal{F}_1(n,m)$ is a real $2m^2-6m+5-\sum^{m-n-1}_{i=1}{m-2 \choose n-1+i}$ dimensional subset of a algebraic variety with the same real dimension when $m>n+1$.

Reversible quaternionic hyperbolic isometries

Let G be a group. An element g∈G is called reversible if it is conjugate to g−1 within G, and called strongly reversible if it is conjugate to g−1 by an order two element of G. Let HHn be the n-dimensional quaternionic hyperbolic space. Let PSp(n,1) be the isometry group of HHn. In this paper, we classify reversible and strongly reversible elements in Sp(n) and Sp(n,1). Also, we prove that all the elements of PSp(n,1) are strongly reversible.

ON DISCRETENESS OF SUBGROUPS OF QUATERNIONIC HYPERBOLIC ISOMETRIES

Let $\mathbf{H}_{\mathbb{H}}^{n}$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group $\text{Sp}(n,1)$ acts on $\mathbf{H}_{\mathbb{H}}^{n}$ by isometries. A subgroup $G$ of $\text{Sp}(n,1)$ is called Zariski dense if it neither fixes a point on $\mathbf{H}_{\mathbb{H}}^{n}\cup \unicode[STIX]{x2202}\mathbf{H}_{\mathbb{H}}^{n}$ nor preserves a totally geodesic subspace of $\mathbf{H}_{\mathbb{H}}^{n}$. We prove that a Zariski dense subgroup $G$ of $\text{Sp}(n,1)$ is discrete if for every loxodromic element $g\in G$ the two-generator subgroup $\langle f,gfg^{-1}\rangle$ is discrete, where the generator $f\in \text{Sp}(n,1)$ is a certain fixed element not necessarily from $G$.

On conjugation orbits of semisimple pairs in rank one

Abstract We consider the Lie groups SU ⁢ ( n , 1 ) {\mathrm{SU}(n,1)} and Sp ⁢ ( n , 1 ) {\mathrm{Sp}(n,1)} that act as isometries of the complex and the quaternionic hyperbolic spaces, respectively. We classify pairs of semisimple elements in Sp ⁢ ( n , 1 ) {\mathrm{Sp}(n,1)} and SU ⁢ ( n , 1 ) {\mathrm{SU}(n,1)} up to conjugacy. This gives local parametrization of the representations ρ in Hom ⁢ ( F 2 , G ) / G {{\mathrm{Hom}}({\mathrm{F}}_{2},G)/G} such that both ρ ⁢ ( x ) {\rho(x)} and ρ ⁢ ( y ) {\rho(y)} are semisimple elements in G, where F 2 = 〈 x , y 〉 {{\mathrm{F}}_{2}=\langle x,y\rangle} , G = Sp ⁢ ( n , 1 ) {G=\mathrm{Sp}(n,1)} or SU ⁢ ( n , 1 ) {\mathrm{SU}(n,1)} . We use the PSp ⁢ ( n , 1 ) {{\mathrm{PSp}}(n,1)} -configuration space M ⁢ ( n , i , m - i ) {{\mathrm{M}}(n,i,m-i)} of ordered m-tuples of distinct points in 𝐇 ℍ n ¯ {\overline{{\mathbf{H}}_{{\mathbb{H}}}^{n}}} , where the first i points in an m-tuple are boundary points, to classify the semisimple pairs. Further, we also classify points on M ⁢ ( n , i , m - i ) {{\mathrm{M}}(n,i,m-i)} .

Deformation Spaces of Discrete Groups of SU(2,1) in Quaternionic Hyperbolic Plane: A Case Study

In this note, we study deformations of discrete and Zariski dense subgroups of in the isometry group of quaternionic hyperbolic space. Specifically, we consider two examples coming from representations of 3-manifold groups (the figure eight knot and Whitehead links complement) and show opposite behaviors: one is not deformable outside U(2, 1), while the other has a big space of deformations in .

Growth gap in hyperbolic groups and amenability

We prove a general version of the amenability conjecture in the unified setting of a Gromov hyperbolic group G acting properly cocompactly either on its Cayley graph, or on a CAT(-1)-space. Namely, for any subgroup H of G, we show that H is co-amenable in G if and only if their exponential growth rates (with respect to the prescribed action) coincide. For this, we prove a quantified, representation-theoretical version of Stadlbauer's amenability criterion for group extensions of a topologically transitive subshift of finite type, in terms of the spectral radii of the classical Ruelle transfer operator and its corresponding extension. As a consequence, we are able to show that, in our enlarged context, there is a gap between the exponential growth rate of a group with Kazhdan's property (T) and the ones of its infinite index subgroups. This also generalizes a well-known theorem of Corlette for lattices of the quaternionic hyperbolic space or the Cayley hyperbolic plane.

Inverse mean curvature flow in quaternionic hyperbolic space

In this paper we complete the study started in [Pi2] of evolution by inverse mean curvature flow of star-shaped hypersurface in non-compact rank one symmetric spaces. We consider the evolution by inverse mean curvature flow of a closed, mean convex and star-shaped hypersurface in the quaternionic hyperbolic space. We prove that the flow is defined for any positive time, the evolving hypersurface stays star-shaped and mean convex. Moreover the induced metric converges, after rescaling, to a conformal multiple of the standard sub-Riemannian metric on the sphere defined on a codimension 3 distribution. Finally we show that there exists a family of examples such that the qc-scalar curvature of this sub-Riemannian limit is not constant.

Characterizing the harmonic manifolds by the eigenfunctions of the Laplacian

The space forms, the complex hyperbolic spaces and the quaternionic hyperbolic spaces are characterized as the harmonic manifolds with specific radial eigenfunctions of the Laplacian.

The Weil–Petersson curvature operator on the universal Teichmüller space

The universal Teichm\"uller space is an infinitely dimensional generalization of the classical Teichm\"uller space of Riemann surfaces. It carries a natural Hilbert structure, on which one can define a natural Riemannian metric, the Weil-Petersson metric. In this paper we investigate the Weil-Petersson Riemannian curvature operator $\tilde{Q}$ of the universal Teichm\"uller space with the Hilbert structure, and prove the following: (i) $\tilde{Q}$ is non-positive definite. (ii) $\tilde{Q}$ is a bounded operator. (iii) $\tilde{Q}$ is not compact; the set of the spectra of $\tilde{Q}$ is not discrete. As an application, we show that neither the Quaternionic hyperbolic space nor the Cayley plane can be totally geodesically immersed in the universal Teichm\"uller space endowed with the Weil-Petersson metric.

Shimizu\u2019s Lemma for Quaternionic Hyperbolic Space

We give a generalisation of Shimizu’s lemma to complex or quaternionic hyperbolic space in any dimension for groups of isometries containing an arbitrary parabolic map. This completes a project begun by Kamiya (Hiroshima Math J 13:501–506, 1983). It generalises earlier work of Kamiya, Inkang Kim and Parker. The analogous result for real hyperbolic space is due to Waterman (Adv Math 101:87–113, 1993).

A remark on a triple points in the boundary of quaternionic hyperbolic space

In this paper we consider a triple of distinct points in the boundary of quaternionic hyperbolic space and detect where these points are by using the quaternionic triple product.