Betti numbers of locally symmetric spaces

Let \( G \) be a semisimple Lie group, for example:

• \( G = \mathrm{SL}_2({\mathbb R}) \) or \( \mathrm{SL}_m({\mathbb R}), m \ge 3 \);
• \( G = \mathrm{SO}(p, q), p, q \ge 1 \) ;
• \( G = \mathrm{SL}_2({\mathbb C}) \), which is isogenic to \( \mathrm{SO}(3,1) \).

Let \( K \) be a maximal compact subgroup of \( G \), and let \( X = G/K \) be the associated Riemannian symmetric space, whose group of orientation-preserving isometries is \( G \) (at least if the latter is connected). In the examples:

• If \( G = \mathrm{SL}_2({\mathbb R}) \), we can take \( K = \mathrm{SO}(2) \) and \( X \) is then the hyperbolic plane \( {\mathbb H}^2 \);
• If \( G = \mathrm{SL}_2({\mathbb C}) \), \( K = \mathrm{SU}(2) \) then \( X \) is the hyperbolic space \( {\mathbb H}^3 \);
• If \( G = \mathrm{SL}_m({\mathbb R}) \) and \( K = \mathrm{SO}(m) \) then \( X \) is the subset of the projective space of m by m matrices which is the image of the cone of positive definite matrices.

A \( X \)-manifold is a complete Riemannian manifold \( M \) in which each point has a neighbourhood isometric to an open subset of \( X \). It can be written as a quotient \( M = \Gamma \backslash X \) where \( \Gamma \) is a discrete, torsion-free subgroup of \( G \) (this being equivalent to it acting freely, properly discontinuously on \( X \)). We want to relate the topological properties of \( M \) with the metric properties of its locally symmetric metric. A first result in this direction is the following.

Theorem (Gromov): For any symmetric space \( X \) as above, there exists a constant \( C_X \) such that for all \( X \)-manifolds \( M \) of finite volume we have :
\[
\sum_{q=1}^{\dim(X)} b_q(M) \le C_X \mathrm{vol}(M).
\]

Conjecturally, the following more powerful relation holds.

Conjecture (Gelander): For any symmetric space \( X \) as above not isometric to hyperbolic space \( {\mathbb H}^3 \), there exists a constant \( \beta(X) \) such that any \( X \)-manifold \( M \) of finite volume is homotopy equivalent to a simplicial complex with at most \( \beta(X) \mathrm{vol}(M) \) simplices.

Remarks :

• Note that the case \( X = {\mathbb H}^3 \) is a genuine exception due to hyperbolic Dehn surgery which shows the existence of infinitely many homotopy types of hyperbolic 3-manifolds with volume below a fixed constant. The conjecture is still expected to hold in this case with the added hypothesis that the manifolds be arithmetic.
• Gelander proved the conjecture restricted to non-compact manifolds.
• In general, the problems in proving the conjecture come from the topology of the compact regions in the thin part, the neighbourhoods of short closed geodesics.

Betti numbers in covers

If we restrict attention to the set of finite covers of a given \( X \)-manifold \( M \) then the conclusion of the conjecture above holds trivially: pulling back a fixed triangulation \( K \) of \( M \) to a finite cover gives triangulations with a number of simplices proportional to the degree of the cover, hence to the volume of the covering space.

On the other hand the Betti numbers are nit so well-behaved as to be multiplicative. It is still possible, in certain situations, to give a better estimate for them than the one from Gromov’s general bounds.

Let
\[
\cdots \overset{d_{q-1}}{\to} C^q(K) \overset{d_q}{\to} C^{q+1}(K) \to \cdots
\]
be the cochain complex of \( K \), and \( Z^q = \ker(d_q)\), \( B^q = \mathrm{Im}(d_{q-1})\) so that the cohomology is \( H^q(M) = Z^q / B^q \). On \(C^q(K) \otimes {\mathbb R} \) consider the inner product for which the base dual to the \( q \)-simplices is orthogonal. Then there is the « Hodge decomposition » as an orthogonal sum:
\[
C^q(K) = \left( \ker(d_q) \cap \ker(d_{q-1}^*) \right) \oplus \mathrm{Im}(d_{q-1}) \oplus \mathrm{Im}(d_q^*)
\]
so that if we denote by \( \mathcal H^q = \ker(d_q) \cap \ker(d_{q-1}^*) \) the space of harmonic cochains there is a linear isomorphism \( H^q(K) \otimes {\mathbb R} \cong \mathcal H^q \).

This can also be formulated in terms of the « Laplace operator » \( \Delta_q = d_{q-1}d_{q-1}^* + d_q^*d_q \), the kernel of which is equal to \( \mathcal H^q \). Thus, if we let \( P_q \) be the orthogonal projection of \( C^q(K) \) onto \( \mathcal H^q \) we have \( b_q(K) = \mathrm{tr}(P_q) \).

Now consider a regular finite cover \( \overline K \to K \) with Galois group \( H \). The Laplacians \( \overline \Delta_q \) and the projections \( \overline P_q \) onto their kernels are \( H \) -equivariant operators. Let \( \sigma_1, \ldots, \sigma_r \) be the \( q \)-simplices of \( K \) and choose lifts \( \bar \sigma_1, \ldots, \bar \sigma_r \) to \( \overline K \). Then we have
\[
\frac 1 {|H|} b_q(\overline K) = \frac 1 {|H|} \sum_{i=1}^r \sum_{h \in H} \langle \overline P_q (h\bar \sigma_i), h\bar \sigma_i \rangle = \sum_{i=1}^r \langle \overline P_q \bar\sigma_i, \bar \sigma_i \rangle.
\]
Define the right-hand side to be \( \mathrm{tr}_H(\overline P_q) \). Note that it makes sense also for an infinite \( H \). In particular, if \( H = \pi_1(K) \) (so that \( \overline K = \widetilde K \) is the universal cover of \( K \)) this defines the \( L^2 \)-Betti numbers of \( K \):
\[
b_q^{(2)}(K) = \sum_{i=1}^r \langle \widetilde P_q \tilde\sigma_i, \tilde\sigma_i \rangle.
\]

Lück Approximation Theorem: Suppose that \( \Gamma_n \) is a sequence of finite-index, normal subgroups in \( \Gamma = \pi_1(K) \) such that \( \Gamma_n \supset \Gamma_{n+1} \) and \( \bigcap_n \Gamma_n = \{1\} \). Then we have for \( K_n = \Gamma_n \backslash \widetilde K \):
\[
\lim_{n \to +\infty} \left( \frac{b_q(K_n)} { |\Gamma / \Gamma_n|} \right) = b_q^{(2)}(K).
\]

Remarks:

• If \( K \) is a triangulation of a locally symmetric manifold of finite volume this is more precise than Gromov’s bound;
• On the other hand, the \( L^2 \)-Betti numbers may a priori depend on \( K \) itself.

Locally homogeneous setting

We now work with \( M = \Gamma \backslash X \) a \( X \)-manifold of finite volume, where \( X = G/K \) is a symmetric space as above. We will use the basepoint \( x_0 = eK \in X \).

To compute the cohomology of \( M \), instead of a cochain complex we can use the de Rham complex of differential forms on \( M \). In our locally homogeneous setting this can be written as:
\[
(\ast) \qquad C^q(M) = \mathrm{Hom}_K \left( \wedge^q\mathfrak p, C^\infty(\Gamma \backslash G) \right)
\]
where \( \mathfrak p = T_{x_0}X \) is the tangent space at \(x_0 \) of \( X \), which may be identified with the quotient space \( \mathfrak G / \mathfrak k \) of tje Lie algebras of \( G \) and \( K \), and is thus a \( K \)-module via the adjoint representation of \( K \) on \( \mathfrak g \). The space \( C^\infty(\Gamma \backslash G) \) is a \( K \)-module via the right-regular representation given by \( kf = f(\cdot k) \).

To get a differential form \( \omega_\phi \) from an element \( \phi \in C^q(M) \) we can compute its action on a given \( w \in \wedge^q M \): choose \( g \in G \) such that \( w = g_*v \) where \( v \in T_{x_0}M = \mathfrak p \) and put:
\[
\omega_\phi(w) = \phi(v)(g).
\]
If we choose another \( g’ \) in the computation then we have \( g’ \in \Gamma g K \) and it follows that this is well-defined.

Note that this setup is roughly equivalent to the discrete case: instead of looking at a set of representatives for the action of a discrete groups on a simplicial complex, we look only at one point since the action of the Lie group \( G \) is transitive.

In the model \( (\ast) \) for the de Rham complex, the differentials are computed as follows:
\[
d\eta(y_0, \ldots, y_q) = \sum_{i=1}^q (-1)^i y_i \cdot \eta(y_0,\ldots,\hat y_i, \ldots, y_q)
\]
(where we view \( y_i \in \mathfrak g \) as a vector field on \( G \) and hence as a differential operator on \( C^\infty(\Gamma \backslash G) \)).

There is a natural inner product on \( C^q(M) \) which is given by identifying it with the space
\[
\left( \wedge^q \mathfrak p^* \otimes C^\infty(\Gamma \backslash G) \right)^K
\].

With respect to the inner product defined above the de Rham differential \( d \) has an adjoint \( d^* \). The Hodge Laplacian on \( C^q(M) \) is the operator \( \Delta_q = d_q^*d_q + d_{q-1}d_{q-1}^* \). The latter can also be described in more algebraic terms: let \( \Omega \) be the Casimir operator of \( G \), the element in the universal enveloping algebra of \( \mathfrak g \) given by \( \Omega = \sum_s x_s x_s’\) where \( x_s \) is a basis of \( \mathfrak g \) and \( x_s’ \) its dual basis with respect to the Killing form.

Kuga’s lemma: Let \( \rho_\Gamma \) be the right regular representation of \( G \) on \( L^2(\Gamma \backslash G ) \). For \( \eta \in C^\infty(\Gamma \backslash G \) we have :
\[
\Delta_q\eta = \rho_\Gamma(\Omega)\eta.
\]

By the Hodge-de Rham theorem the Betti number \( b_q(M) \) is equal to the dimension of the kernel of the operator \( \Delta_q \), or to the trace of the orthogonal projection \( P_q \) onto this kernel. A convenient analytic way to express this dimension is via the heat operator \( e^{-t\Delta_q} \). From the definition we see that for any square-integrable \( q \)-form \( \eta \) we have
\[
P_q \eta = \lim_{t \to +\infty} e^{-t\Delta_q}\eta.
\]
The heat operator has a kernel \( e^{-t\Delta_q}(\cdot, \cdot) \) (the heat kernel), which is an element of the space
\[
\left( \mathrm{End}(\wedge^q \mathfrak p^*) \otimes C^\infty(\Gamma \backslash G \times \Gamma \backslash G) \right)^{K \times K}
\]
so that \( (e^{-t\Delta}\eta)_x = \int_M e^{-t\Delta(x,y)}\eta_y dy \). The Betti numbers can then be computed via the formula:
\[
b_q(M) = \lim_{t \to +\infty} \int_M \mathrm{tr}(e^{-t\Delta_q}(x, x)) dx.
\]