Higher Rank Symmetric Spaces Research Articles

Growth of mod-2 homology in higher-rank locally symmetric spaces

Let X be a higher-rank symmetric space or a Bruhat–Tits building of dimension at least 2 such that the isometry group of X has property (T). We prove that for every torsion-free lattice Γ⊂IsomX, any homology class in H1(Γ∖X,F2) admits a representative cycle of total length oX(Vol(Γ∖X)). As an application, we show that dimF2H1(Γ∖X,F2)=oX(Vol(Γ∖X)).

An Area Theorem for Joint Harmonic Functions on the Product of Homogeneous Trees

For harmonic functions v on the disc, it has been known for a long time that non-tangential boundedness a.e.is equivalent to finiteness a.e. of the integral of the area function of v (Lusin area theorem). This result also hold for functions that are non-tangentially bounded only in a measurable subset of the boundary, and has been extended to rank-one hyperbolic spaces, and also to infinite trees (homogeneous or not). No equivalent of the Lusin area theorem is known on higher rank symmetric spaces, with the exception of the degenerate higher rank case given by the cartesian product of rank-one hyperbolic spaces. Indeed, for products of two discs, an area theorem for jointly harmonic functions was proved by M.P. and P. Malliavin, who introduced a new area function; non-tangential boundedness a.e. is a sufficient condition, but not necessary, for the finiteness of this area integral. Their result was later extended to general products of rank-one hyperbolic spaces by Korányi and Putz. Here we prove an area theorem for jointly harmonic functions on the product of a finite number of infinite homogeneous trees; for the sake of simplicity, we give the proofs for the product of two trees. This could be the first step to an area theorem for Bruhat–Tits affine buildings, thereby shedding light on the higher rank continuous set-up.

Minakshisundaram–Pleijel coefficients for non-compact higher rank symmetric spaces

We assign to a non-compact Riemannian symmetric space X a theta function and a zeta function. We compute all of the Minakshisundaram–Pleijel coefficients in a short-time asymptotic expansion of the theta function especially when X is of complex type, which we use to compute the one-loop effective potential—whose relevance for quantum field theory, for example, is briefly commented on. These coefficients, for X of general type, are also shown to be specifically related to residues and special values of the zeta function. The results presented extend previous ones in the literature where the rank of X is assumed to be one.

A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries

Abstract We prove the following rank rigidity result for proper CAT ⁡ ( 0 ) {\operatorname{CAT}(0)} spaces with one-dimensional Tits boundaries: Let Γ be a group acting properly discontinuously, cocompactly, and by isometries on such a space X. If the Tits diameter of ∂ ⁡ X {\partial X} equals π and Γ does not act minimally on ∂ ⁡ X {\partial X} , then ∂ ⁡ X {\partial X} is a spherical building or a spherical join. If X is also geodesically complete, then X is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of ∂ ⁡ X {\partial X} , does not require the Tits diameter to be π, and we give an alternate condition that guarantees rigidity when this hypothesis is removed, which is that a certain invariant of the group action be even.

Relations Between Laplace Spectra and Geometric Quantization of Reimannian Symmetric Spaces

We consider a modified Kostant-Souriau geometric quantization scheme due to Czyz and Hess for Hamiltonian systems on the cotangent bundles of compact rank-one Riemannian symmetric spaces (CROSS). It is used, together with a symplectic reduction process, to relate its energy spectrum to the spectrum of the Laplace-Beltrami operator. Moreover, the corresponding eigenspaces have real dimension equal to the complex dimension of the space of the holomorphic sections of the quantum bundle which is obtained after the quantization. The relation between the two constructions was first noticed by Mladenov and Tsanov for the case of the spheres. In addition to the CROSS case, we announce preliminary results related to the case of compact Riemannian symmetric spaces of higher rank.

Simplicial volume, barycenter method, and bounded cohomology

We prove that the locally finite simplicial volume of most \({\mathbb {Q}}\)-rank 1 locally symmetric spaces is positive, which has been open for many years. As a corollary, we improve the degree theorem of Connell and Farb for \({\mathbb {Q}}\)-rank 1 locally symmetric spaces. During the course of a proof, we show that codimension one dimensional Jacobian of the barycentric straightening map is uniformly bounded for most higher rank symmetric spaces. We also address the issue of surjectivity of the comparison map for semisimple Lie groups of rank 2.

Limit sets and convex cocompact groups in higher rank symmetric spaces

Limit sets and convex cocompact groups in higher rank symmetric spaces

The spectrum and heat dynamics of locally symmetric spaces of higher rank

The aim of this paper is to study the spectrum of the$L^{p}$Laplacian and the dynamics of the$L^{p}$heat semigroup on non-compact locally symmetric spaces of higher rank. Our work here generalizes previously obtained results in the setting of locally symmetric spaces of rank one to higher rank spaces. Similarly as in the rank-one case, it turns out that the$L^{p}$heat semigroup on$M$has a certain chaotic behavior if$p\in (1,2)$, whereas for$p\geq 2$such chaotic behavior never occurs.

Polar actions on symmetric spaces of higher rank

We show that polar actions of cohomogeneity 2 on simple compact Lie groups of higher rank, endowed with a biinvariant Riemannian metric, are hyperpolar. Combining this with a recent result of the second-named author, we are able to prove that polar actions induced by reductive algebraic subgroups in the isometry group of an irreducible Riemannian symmetric space of higher rank are hyperpolar. In particular, this result affirmatively settles the conjecture that polar actions on irreducible compact symmetric spaces of higher rank are hyperpolar.

Some questions on integral geometry on noncompact symmetric spaces of higher rank

Let \(M\) be a noncompact symmetric space of higher rank. We consider two types of averages of functions: one, over level sets of the heat kernel on \(M\) and the other, over geodesic spheres. We prove injectivity results for functions in \(L^p\) which extend the results in Pati and Sitaram (Sankya Ser A 62:419–424, 2000).

Cross-ratio in higher rank symmetric spaces

Cross-ratio in higher rank symmetric spaces

Symmetric spaces of higher rank do not admit differentiable compactifications

Any nonpositively curved symmetric space admits a topological compactification, namely the Hadamard compactification. For rank one spaces, this topological compactification can be endowed with a differentiable structure such that the action of the isometry group is differentiable. Moreover, the restriction of the action on the boundary leads to a flat model for some geometry (conformal, CR or quaternionic CR depending of the space). One can ask whether such a differentiable compactification exists for higher rank spaces, hopefully leading to some knew geometry to explore. In this paper we answer negatively.

Relative rigidity, quasiconvexity andC–complexes

We introduce and study the notion of relative rigidity for pairs $(X,\JJ)$ where 1) $X$ is a hyperbolic metric space and $\JJ$ a collection of quasiconvex sets 2) $X$ is a relatively hyperbolic group and $\JJ$ the collection of parabolics 3) $X$ is a higher rank symmetric space and $\JJ$ an equivariant collection of maximal flats Relative rigidity can roughly be described as upgrading a uniformly proper map between two such $\JJ$'s to a quasi-isometry between the corresponding $X$'s. A related notion is that of a $C$-complex which is the adaptation of a Tits complex to this context. We prove the relative rigidity of the collection of pairs $(X, \JJ)$ as above. This generalises a result of Schwarz for symmetric patterns of geodesics in hyperbolic space. We show that a uniformly proper map induces an isomorphism of the corresponding $C$-complexes. We also give a couple of characterizations of quasiconvexity. of subgroups of hyperbolic groups on the way.

Almost-minimal nonuniform lattices of higher rank

If is a nonuniform, irreducible lattice in a semisimple Lie group G, with R-rank G ≥ 2, we show contains a subgroup that is isomorphic to a nonuni form, irreducible lattice in either SL3(R), SL3(C), or a direct product SL2(R) m × SL2(C) n , with m + n ≥ 2. (In geometric terms, this can be interpreted as a statement about the existence of totally geodesic subspaces of finite-volume, noncompact, locally symmetric spaces of higher rank.) Another formulation of the result states that if G is any isotropic, almost simple algebraic group over Q, such that R-rankG ≥ 2, then G contains an isotropic, almost simple Q- subgroup H, such that H is quasisplit, and R-rank H ≥ 2.

Geometry and dynamics of discrete isometry groups of higher rank symmetric spaces

For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For symmetric spaces of higher rank, which contain isometrically embedded Euclidean planes, the situation becomes far more complicated. This paper is devoted to the study of the geometric limit set of “nonelementary” discrete isometry groups of higher rank symmetric spaces. We obtain the natural generalisations of some well-known results from Kleinian group theory. Our main tool consists in a detailed description of the dynamics of individual isometries. As a by-product, we give a new geometric construction of free isometry groups with parabolic elements in higher rank symmetric spaces.

The uniqueness of the maximal measure for geodesic flows on symmetric spaces of higher rank

In this paper we show that the geodesic flow on a compact locally symmetric space of nonpositive curvature has a unique invariant measure of maximal entropy. As an application to dynamics we show that closed geodesics are uniformly distributed with respect to this measure. Furthermore, we prove that the volume entropy is minimized at a compact locally symmetric space of nonpositive curvature among all conformally equivalent metrics with the same total volume.

Spherical rank rigidity and Blaschke manifolds

In this paper we give a characterization of locally compact rank one symmetric spaces, which can be seen as an analogue of Ballmann's and Burns and Spatzier's characterizations of nonpositively curved symmetric spaces of higher rank, as well as of Hamenstädt's characterization of negatively curved symmetric spaces. Namely, we show that a complete Riemannian manifold M is locally isometric to a compact, rank one symmetric space if M has sectional curvature at most 1 and each normal geodesic in M has a conjugate point at π .

Measures on the geometric limit set in higher rank symmetric spaces

Measures on the geometric limit set in higher rank symmetric spaces

Asymptotic Dirichlet Problem for Harmonic Maps with Bounded Image

Existence and uniqueness is proved for asymptotic Dirichlet problems on Hadamard manifolds. This includes manifolds of bounded negative curvature and symmetric spaces of higher rank.

Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces

We compute the quasi-isometry group of an irreducible nonuniform lattice in a semisimple Lie group with finite center and no rank one factors, and show that any two such lattices are quasi-isometric if and only if they are commensurable up to conjugation.