The rigid world of symmetric spaces of noncompact type

The current post is based on a talk I gave at the university of Leiden which unfortunately I completely butchered due, in part, to the sudden realization that I had not chance to get through this material in the given time. Nothing here is new or proved, though some sketches are added when I find it relevant.

The (unattained) goal of the presentation was to show how much of the geometry of a symmetric space of noncompact type can be determined from it boundary sphere alone (in rank 1 cases) and a simplicial complex called the boundary at infinity (in higher rank cases) by following the evolution of the Mostow rigidity theorem from its origins (closed quotients of real hyperbolic spaces) to higher ranks.

The next installment of these notes will be to look at cohomological rigidity of these spaces. A lot can actually be said about the group cohomology of the fundamenta group of a locally symmetric space of noncompact type by looking at its action on the boundary sphere of the space. There are also interesting KK-cycles associated with the buildings at infinity.

Basic definitions

A map $f_p: X\to X$ is said to be a geodesic symmetry if it fixes $p$ and reverses all geodesics centered at $p\in X$ . A complete Riemannian space is called (locally) symmetric if the geodesic symmetries are (local) isometries.
Any simply connected locally symmetric space is a symmetric space, hence any locally symmetric space $X$ is a quotient of a symmetric space (its universal cover) by the action of its fundamental group.

The rank of the symmetric space is the largest dimension of its flat totally geodesic submanifolds.

For any symmetric space $X$ the group $G = Iso(X)^0$ acts transitively (proof – take any two points $x, y\in X$ and the unique geodesic connecting them $\gamma(0) =x$ , $\gamma(t_0) = y$ , then the geodesic symmetry $f_{\gamma(t_{0}/2)}$ sends $x$ to $y$ ).
It follows that all symmetric spaces are homogeneous spaces – if $x\in X$ , the group $K = Stab(x)^0$ is compact subgroup in $G$ , and

$G/K \simeq X.$

Types of symmetric spaces

Euclidean type (zero curvature)
Compact type (non-negative curvature)
Noncompact type (non-positve curvature)

In the above definitions, to make the types distinct, one assumes that the compact and noncompact type do not have any euclidean factors in their de Rham decomposition. For instance, $\H_\R^n$ is a symmetric space of noncompact type, $\H_\R^n\times \R$ is not, since it contains a euclidean factor $\R$ (though the curvature is still non-positive).

As the above example suggests, symmetric spaces are not all of the above types, but their de Rham factors will be of these types. Recall that the de Rham factors are the submanifolds corresponding to integral subbundles of the tangent bundle that are also irreducible modules under the action of the monodromy group.

Note also that the curvature need of course not be constant since the isometry group need not act transitively on the Grassmannian of 2-planes $Gr(2, T_pX)$ space (a weaker condition than 2-transitivity), but things like scalar curvature will of course be constant for any symmetric space. Spaces where the isometry group acts transitively on the tangent space at each point are called isotropic manifolds, $S^n$ are examples of such spaces. Noncompact type symmetric spaces are isotropic if and only if their rank is 1.

The geodesic compactification

Given a non-positvely curved symmetric Riemannian space $X$ (or hyperbolic length space) the geodesic compactification of $X$ is defined as follows:

Two geodesics $\gamma, \gamma'$ are said to be asymptotically equivalent if

$\sup_{t\geq 0} d(\gamma(t), \gamma'(t)) < \infty.$

Denote by $[\gamma]$ this equivalence class and set

$\partial X = \{ [\gamma]~|~ \gamma \text{ unit speed geodesic } \}.$

Fixing $x_0 \in X$ we have the following lemma

Lemma 1 – Each $[\gamma]$ has a unique representative centered at $x_0$ , and $\partial X\simeq S^{n-1} \subset T_{x_0}X$ as the unit tangent sphere at some (any) fixed point in $X$ .

The space $\overline{X} = X\cup \partial X$ endowed with a certain topology called the conic topology is a compact space called the geodesic, or visual, or Gromov compactification of $X$ . The conic topology is the topology induced by the open sets in $X$ and for each $[\gamma]\in \partial X$ with $\gamma(0) = x_0$ , the set of truncated open cones of geodesic rays from $x_0$ containing $\gamma(\R^+)$ is a neighborhood basis at $[\gamma]$ (see for instance ). Exercise – Convince yourself that the points $(\gamma(n))_{n \in \N}$ converge to $[\gamma]$ in this topology.

The action of $G$ extends to an action by homeomorphisms on this compactified space by

$g[\gamma] = [g\circ \gamma]$

Under the identification in Lemma 1 above, the action of $K= Stab(x_0)$ is just the induced action of $K$ on the tangent space $T_{x_0}X$ , given by

$k\cdot v= (dk)v$

where $dk$ is the differential of $k$ at $x_0$ . The action of $G$ is usually defined as the adjoint action by setting $x_0 = e$ (the identity in $G$ ) and letting $G$ act by the restriction of the adjoint representation.

The rank 1 case

Here is an exhaustive list of all symmetric spaces of noncompact type of rank 1 –

$\H^n_\R$ ( $\sigma = -1$ )
$\H^n_\C$ ( $-1 \leq \sigma \leq -1/4$ )
$\H^n_\H$ ( $-1 \leq \sigma \leq -1/4$ ) (quaternionic)
(Cayley plane) ( $-1 \leq \sigma \leq -1/4$ )

As can be seen, they consist of real, complex and quaternionic hyperbolic spaces, together with an “exceptional” space called the Cayley plane (or the octonionic hyperbolic plane). The iniquitous looking space $\H^2_\R$ actually sticks out like a sore thumb with pathological behaviour that is not exhibited by other spaces (for an example see this post) with loads of non-arithmetic lattices and lack of rigidity. See the very well written article for explicit realizations of these spaces as quotients of Lie groups.

Here are some properties of the compactification $\overline{X}$ when $X$ is any of the above spaces:

Transitive boundary action: The $G$ -action on this compactification is transitive on the boundary $\partial X$ .
Parabolic subgroups: There is a 1-1 correspondence between parabolic subgroups of $G$ and points on $\partial X$ given by $\partial X\ni x \mapsto Stab(x)\subset G$ , hence there is only one conjugacy class of parabolic subgroups.
Harmonic densities (see ) – There is a family of probability measures on parametrized by satisfying the following properties
1. The map $x\mapsto \mu_x$ is continuous (with respect to the weak topology on the measures on $\partial X$ ) and $G$ -equivariant.
2. The Radon-Nikodym derivative is
  $\frac{d\mu_x}{d\mu_0}(s) = e^{-r b_s(x)}$
  where $b_s(x) = \lim d(x, s(t))- t$ is called the Busemann function (normalized at $s(0)$ , meaning $b_s(s(0)) = 0$ ) and $r = dim(\partial X)$ . For the Poincare disk $e^{-r b_s(x)} = P(x, s)$ is the Poisson kernel $P(x, s) = \frac{1 - |x|^2}{|x-s|^2}$ .
3. If $x_i\in X$ and $x_i\to x \in \partial X$ , then
  $\mu_{x_i}\to \delta_x \qquad \text{weakly}$

These are some of the general features of rank 1 symmetric spaces. Let’s look at applications of these boundary measures:

Example 1: Equivariant splitings

The compactification above gives us a natural short exact sequence of C*-algebras

$0\to C_0(X) \to C(\overline{X}) \to C(\partial X) \to 0$

given by the assignment

$C(\partial X)\ni f \mapsto F_f \in C(\overline{X})\qquad F_f(x) = \begin{cases} f(x) & x\in \partial X \\ \int_{\partial X} f(v) d\mu_x(v)\end{cases}.$

The properties listed above ensures that the map is equivariant and well defined. The above also works for real valued function algebras of course. This technique is used in to produce concrete realizations of KK-cycles corresponding to the extension.

Example 2: Classical harmonic analysis

For the disk the splitting in the previous example determines a bijection between $C(\partial X)$ and the bounded harmonic measures on $X$ . This is the famous Poisson integral formula of bounded harmonic functions on the disk.

Example 3: Barycenter extension method

Yet another important application of the harmonic densities is the following technique used in the proofs of various classical results (like the Mostow rigidity theorems and the Entropy Rigidity Conjecture for rank 1 symmetric spaces). See for instance \cite. Succinctly it allows one to create a map between two symmetric spaces from a continuous map of their boundaries.

For any measure $\lambda$ in the Lebesgue class on the boundary $\partial X$ we can assign to it a point in $X$ called the barycenter of $\lambda$ defined to be the minimum of the function –

$bar(\lambda) = min_{x\in X}\left\{ x\to \int_{\partial X} b_s(x) d\lambda(s)\right\}.$

This minimum exists since the function $x\mapsto \int_{\partial X} b_s(x) d\lambda(s)$ is strictly convex on $X$ . Given any continuous map

$f: \partial X\to \partial Y$

we can produce a map by the following assignment

$F: X\to Y \qquad F(x) = bar(f_*\mu_x)$

which in many cases turns out to be an isometry. Using this method one can create smooth representative maps $F: X\to Y$ in every homotopy class (where $X$ , $Y$ are negatively curved closed locally symmetric spaces).

Let’s take a quick look at Mostow’s rigidity theorem: The (original) Mostow rigidity theorem states that every closed locally symmetric space of constant curvature $-1$ and dimension $\geq 3$ is uniquely determined up to isometry by its fundamental group. The theorem has been extended to higher ranks and closedness has been replaced by finite volume.

A sketch of the proof goes as follows: Let $X$ , $Y$ be as above. Since $X$ and $Y$ are aspherical manifolds (or $K(\Gamma, 1)$ -spaces, where $\Gamma= \pi_1(X) = \pi_1(Y)$ ) there exists a homotopy equivalence $f: X\to Y$ . This lifts to a $\Gamma$ -equivariant continuous map $\tilde{f}: \H^n\to \H^n$ , which can be chosen to be a quasi-isometry (we need compactness for this).

Quasi-isometries preserve the equivalence relations $\gamma \sim \gamma'$ on the space of geodesics defined earlier, hence induce a $\Gamma$ -equivariant homeomorphism $\partial \tilde{f}: \partial \H^n\to \partial \H^n$ .

Let $\tilde{F}(x) = bar(f_\star \mu_x)$ . This map is $\Gamma$ -equivariant and so descends to an isometry of the quotient space

$F: X\to Y$

we need $dim(X)\geq 3$ for this. There are many known counter examples of this theorem in dimension $2$ .

The higher rank case

A positive Weyl chamber $W_P$ on the boundary $\partial X$ associated with a minimal parabolic subgroup $P\subset G$ is defined to be

$W_P = \{ x\in \partial X ~|~ Stab(x) = P\}.$

If $X$ is of rank 1, this would be a single point, but for higher rank case this is never the case. The image of a maximal flat subspace $A\subset X$ projected onto the boundary $\partial X$ is tesselated by the closure of a finite family of positive Weyl chambers. Similarly, the whole boundary is tesselated by the the collection of (closed) positive Weyl chambers.

Here are several strange features/pathologies of symmetric spaces of higher rank:

Parabolic subgroups: There is no longer a 1-1 correspondence between parabolic subgroups of $G$ and points on $\partial X$ . The correspondence $x\mapsto Stab(x)$ does only map surjectively onto the collection of parabolic subgroups.
No transitive boundary action: The $G$ -orbits on the boundary are parametrized by any fixed (closed) positive Weyl chamber, i.e. $G \backslash \partial X = \overline{W_P}$ for any minimal parabolic subgroup $P$ .
Lack of a smooth structure: There is no natural smooth structure on $\overline{X}$ compatible with that of $X$ making $G$ act by smooth maps (see ). It can be thought of as a smooth manifold with a finte set of signularities (one for each conjugacy class of maximal parabolic subgroups) (see ).
No harmonic densities: The methods that created the harmonic densities, now only produce a family of measures $(\mu_x)_{x\in X}$ supported on a single $G$ -orbit on $\partial X$ (see ).

In the rank 1 case, the geodesic boundary in a sense played the role as an isomorphism invariant, as we saw in the proof of the original Mostow rigidity theorem. The new construction needed as our isomorphism invariant is the building at infinity or Tits building. Let’s look at what it is –

The collection of all closed positive Weyl chambers have a natural structure as an abstract simplicial complex called a thick spherical Tits building (or simply a building) often refered to as the building at infinity. A building is simplicial complex where the maximal simplices are all of the same dimension (these will be called chambers together with a family of subcomplexes called apartments endowed with a group action of a reflection group (such complexes are called Coxeter complexes). The word spherical indicates that the reflection group of each apartment is finite, while the word thick means that each simplex of codimension 1 is contained in at least 3 chambers.

Building at infinity –

A simplicial complex $\Delta$ is called a (thick) spherical Tits building if it contains a family of subsets called apartments that satisfy the following conditions:

Any two simplices are contained in some apartment
Any two chambers (maximal simplices) are of the same dimension.
Any two chambers in the same apartment are connected by a finite sequence of adjacent chambers (i.e. chambers sharing a face).
There is a finite group of “reflections” acting on each apartment, such that the action is simply transitive on each chamber (called a finite Coxeter complex).
For any two simplices $\sigma, \sigma'$ contained in the intersection $\Sigma \cap \Sigma'$ of two apartments, there is a simplicial isomorphism $\Sigma \to \Sigma'$ fixing $\sigma$ and $\sigma'$ .
(Thick) – Any wall of a chamber is contained in at least $3$ chambers

The collection of all positive (closed) Weyl chambers form a thick spherical Tits building, with apartments given by the images of maximal flats onto the boundary sphere, chambers given by the positive Weyl chambers, reflection groups given by the “Weyl groups” of the flat $A$ which are defined to be $W_A = N(A)/Z(A)$ (normalizer over centralizer of $A$ ).\footnote{As an abstract group $W_A$ is independent of choice of $A$ and is given as the group of reflections along hyperplanes given by the kernels of the restricted roots in $\mathfrak{a}$ (the Lie algebra of $A$ )}

The reason we care about buildings is the following reformulation of a deep theorem of Tits

Theorem 1 – For any symmetric spaces of non-compact type or rank $\geq 2$ , the space is uniquely determined up to isometry by the isomorphism type of its spherical Tits building at infinity.

To emphasise the point, this simplicial complex we call a building at infinity is entirely created from the boundary data of $\partial X$ (well, also how the maximal flats imbed into $X$ , as this determines the apartments) and the induced group action. With this more rudimentary structure one is still able to pinpoint the underlying space $X$ up to isometric isomorphism.

If follows from the fact that the Weyl chambers form a Tits building that we have –
\begin{proposition}

Proposition 1 – Any two adjacent positive Weyl chambers are contained in the image of one maximal flat totally geodesic submanifold.

From this one readily sees what is the issue with trying to define a smooth (or even $C^k$ for $k\geq 1$ ) structure on $X$ in higher rank. Let’s look at a concrete example, say $G= SL_2(\R)\times \R$ . This is a symmetric space of non-positive curvature, hence a geodesic compactification can be defined. Its geodesic boundary $\partial X$ can be depicted as a sphere $S^2$ . The (closure of the) positive Weyl chambers are all the lines from the north to the south pole tracing out one half of some great circle. The closure of of any of these chambers intersect any of the other, since they all contain the north and the south pole, hence the properties of the building ensures us that we may take three positive Weyl chambers $W_1, W_2, W_3$ (half circles in $\partial X$ ) two maximal flats $A_1$ and $A_2$ in $X$ (determining our apartments, which are now loops in $\partial X$ ) such that their image on $\partial X$ (the apartments) satisfy

$W_1, W_2\subset A_1 \qquad W_1, W_3 \subset A_2$

It should be clear that if $A_1$ is embedded smoothly, then $A_2$ has a cusp at the north and south pole (and vice versa). This line of reasoning is what is used in k where a precise proof can be found.

As for the lack of a transitive boundary action, the intuitive picture is the following, if $\gamma$ is a geodesic contained in precisely $r$ maximal flats $A_1, …, A_r$ , that is $\gamma(\R)\subset A_1\cap …\cap A_r$ , then $g\cdot \gamma$ is contained in precisely $r$ maximal flats $g(A_i)$ as well. So the number of maximal flats containing $\gamma$ remains invariant under the action of $G$ . It can also be invariant under asymptotic equivalence. Hence itwould be impossible to go from one point on the boundary $[\gamma]$ represented by a geodesic contained only in $1$ maximal flat (these are called the regular boundary points) to a point $[\gamma']$ with $\gamma'$ contained in more than $1$ maximal flat with an isometry.

Let’s end with an application of the the building at infinity

Mostow’s rigidity theorem

The building at infinity allows one to extend Mostows rigidity theorem to the case of closed symmetric spaces of noncompact type of rank $\geq 2$ , so these are again uniquely determined by their fundamental groups.

The proof runs parallel to the one stated earlier. Just as in the previous case we have a $\Gamma = \pi_1(X) = \pi_1(Y)$ -equivariant quasi-isometry

$f: \tilde{X}\to \tilde{Y}$

of the respective universal covers of the two closed locally symmetric spaces $X$ and $Y$ . This is shown in to induce an isomorphism of the building at infinity, which descends to a $\Gamma$ -equivariant isometry of the quotients by $\Gamma$ (i.e. the locally symmetric spaces $X$ and $Y$ ). See the cited reference for the proof.

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