Betti numbers of symmetric spaces (continuing previous lecture)

Recall the setup: \( G, K, X=G/K \) are respectively a semisimple Lie group, a maximal compact subgroup and the associated symmetric space; \( \Gamma, M = \Gamma \backslash S \) are a discrete, torsion-free subgroup and the associated \( X \)-manifold. The de Rham complex of \( M \) can be computed via:
\[
C^q(M) = \mathrm{Hom}_K \left( \wedge^q\mathfrak p, C^\infty(\Gamma \backslash G) \right)
\]
and there is a Laplace operator \( \Delta_q \) which is essentially auto-adjoint and positive on the \( L^2 \)-completion of \( C^q \), and an associated heat operator which is given by convolution with the heat kernel
\[
e^{-t\Delta_q} \in \left( \mathrm{End}(\wedge^q \mathfrak p^*) \otimes C^\infty(\Gamma \backslash G \times \Gamma \backslash G) \right)^{K \times K}.
\]
The Betti numbers of \( M \) are then given by the formula:
\[
b_q(M) = \lim_{t \to +\infty} \int_M \mathrm{tr}(e^{-t\Delta_q}(x, x)) dx.
\]

We can make similar definitions to compute the \( L^2 \)-Betti numbers of \( M \). On the space
\[
C_{(2)}^q(X) = \left( \wedge^q \mathfrak p^* \otimes L^2(G) \right)^K
\]
of square-integrable \( q \)-forms on \( X \) there is a Laplace operator \( \Delta_q^{(2)} \) (given by the Casimir action) and an associated heat kernel \( e^{-t\Delta_q^{(2)}} \). On the other hand, since the Laplacian commutes with the left \( G \)-action on \( L^2(G) \) the heat kernel is equivariant with respect to this action, that is :
\[
\forall x,y \in X, \, g \in G : e^{-t\Delta_q^{(2)}}(gx, gy) = g_* e^{-t\Delta_q^{(2)}}(x, y) g^*
\]
hence it is enough to study the following function on \( G \):
\[
k_t(g) = e^{-t\Delta_q^{(2)}}(x_0, gx_0) \in \left( \mathrm{End}(\wedge^q \mathfrak p^*) \otimes \mathscr C(G) \right)^K
\]
(here \( \mathscr C(G) \) is Harish-Chandra’s Schwartz space, a subspace of smooth function very rapidly decreasing at infinity).

The analytic definition for the \( L^2 \)-Betti numbers of \( M \) then goes as follows: choose a fundamental domain \( \mathcal F \) for the action of \( \Gamma \) on \( X \) and put:
\[
b_q^{(2)}(X; \Gamma) = \lim_{t \to +\infty} \int_{\mathcal F} \mathrm{tr}(e^{-t\Delta_q^{(2)}} (x, x)) dx.
\]
This is equal to the combinatorial \( L^2 \)-Betti numbers defined in the previous lecture by a generalisation of the Hodge–de Rham theorem. On the other hand, the equivariance of the \( L^2 \)-heat kernel with respect to the \( G \)-action implies in particular that \( \mathrm{tr}(e^{-t\Delta_q^{(2)}}(gx_0, gx_0) = \mathrm{tr}(e^{-t\Delta_q^{(2)}}(x_0, x_0)) \) for any \( g \in G \), so that:
\[
b_q^{(2)}(X; \Gamma) = \mathrm{vol}(\mathcal F)\lim_{t\to +\infty} \mathrm{tr}(e^{-t\Delta_q^{(2)}}(x_0, x_0)).
\]
Hence all \( L^2 \)-Betti numbers of \( X \)-manifolds are proportional to their volume, with the proportionality coefficient being given by:
\[
\beta_q^{(2)}(X) := \lim_{t\to +\infty} \mathrm{tr}(k_t(e)),
\]
the q-th \( L^2 \)-Betti number of \( X \).

This analytic setting can be used to prove a « uniform » version of the Lück Approximation theorem, dealing with sequences of lattices in \( G \) instead of finite-index subgroups in a single lattice. The basic ingredient is the following expression for the heat kernel on a \( X \)-manifold:
\[
e^{-t\Delta_q}(x, y) = \sum_{\gamma \in \Gamma} e^{-t\Delta_q^{(2)}}(x, \gamma y)
\]
which yields:
\[
b_q(M) = \mathrm{vol}(M) \beta_q^{(2)}(X) + \lim_{t \to +\infty} \int_M \sum_{\gamma \in \Gamma \setminus \{e\}} \mathrm{tr}(e^{-t\Delta_q^{(2)}}(x, \gamma x)) dx.
\]
The second term can be seen to be an \( o(\mathrm{vol}(M)) \) in any sequence with injectivity radius going to infinity, and so we get the following result:
\[
\lim_{\mathrm{inj}(M) \to +\infty} \frac{b_q(M)}{\mathrm{vol}(M)} = \beta_q^{(2)}(X).
\]
Using similar ideas and rigidity results it is possible in certain cases to prove a much stronger result.

Theorem (Abért–Bergeron–Biringer–Gelander–Nikolov–Raimbault–Samet): Let \( G \) be a semisimple Lie group of reak rank at least 2, and with Kazhdan’s property (T). Then for any sequence of lattices \( \Gamma_n \) in \( G \) such that:

• \( \Gamma_n \) is cocompact ;
• The injectivity radius of the \( X \)-manifolds \( \Gamma_n \backslash X \) is uniformly bounded away from 0;
• \( \mathrm{vol}(\Gamma_n \backslash X ) \to +\infty \);

we have
\[
\lim_{n\to+\infty} \frac{b_q(\Gamma_n)}{\mathrm{vol}(\Gamma_n \backslash X)} = \beta_q^{(2)}(X).
\]

Remarks:

• According to Matsushima’s formula the Betti numbers can be written as
  \[
  b_q(M) = \sum_{\Omega(\pi) = 0} m(\pi, \Gamma) \dim \mathrm{Hom}_K(\wedge^q \mathfrak p, \mathcal H_\pi )
  \]
  and it is in fact possible to prove a « limit multiplicity » result about the \( m(\pi, \Gamma) \) themselves, namely:
  \[
  \forall \pi \in \widehat G, \lim_{n \to +\infty} \frac{m(\pi, \Gamma_n)}{\mathrm{vol}(\Gamma_n \backslash X)} = d(\pi)
  \]
  where \( d(\pi) \) is the « formal degree » of \( \pi \). This implies the result about Betti numbers since the \( L^2 \)-Betti numbers are given by:
  \[
  \beta_q^{(2)}(X) = \sum_{\Omega(\pi) = 0} d(\pi) \dim \mathrm{Hom}_K(\wedge^q \mathfrak p, \mathcal H_\pi ).
  \]
• The theorem above is no longer valid for lattices in rank 1 groups, but the conclusion still stands if one adds the condition that the sequence \( \Gamma_n \backslash X \) is « Benjamini–Schramm convergent to \( X \) », meaning that the volume of the R-thin part is negligible before the total volume for any R.

Torsion

A « trivial » bound for torsion homology

Let \( A \) be a free \( {\mathbb Z} \)-module of finite rank, and suppose that the real vector space \( A_{\mathbb R} = A \otimes {\mathbb R} \) is endowed with an inner product. Define \( \mathrm{vol}(A) = \mathrm{vol}(A_{\mathbb R}/A) \) to be the Euclidean volume of a fundamental parallelogram for \( A \).

Now suppose \( f: {\mathbb Z}^a \to {\mathbb Z}^b \) is a morphism; both sides are endowed with a canonical inner product and every submodule is endowed with the induced product. Define the « determinant » \( \det'(f) \) to be the product of all positive singular values of \( f \) (the square roots of positive eigenvalues of the nonegative symmetric operator \( f^*f \)).

Metric rank formula: \( \det'(f) = \mathrm{vol}(\mathrm{Im}(f)) \cdot \mathrm{vol}(\ker(f) \)

As a consequence we get the following bound for the torsion in a finitely generated Abelian group.

Lemma: Let \( Q = \mathrm{coker}(f) \) be a finitely generated Abelian group, \( Q_{\mathrm{free}} \) a maximal free abelian subgroup (which we consider as a lattice in the orthogonal of \( \mathrm{Im}(f) \)) and \( Q_{\mathrm{tors}} \) its torsion subgroup. Then:
\[
| Q_{\mathrm{tors}} | = \det{}'(f) \mathrm{vol}(Q_{\mathrm{free}}) / \mathrm{vol}(\ker(f)) \le \det{}'(f).
\]

If \( C^*, d_* \) is the cochain complex of a simplicial complex \( K \), applying this lemma to \( Q = C^q / \mathrm{Im}(d_{q-1}) \) and noting that \( H^q(K) \) embeds into the latter we get that :
\[
| H^q(K)_{\mathrm{tors}} | \le \| d_{q-1} \|^{\dim(C^q)}
\]
(where \( \| \cdot \| \) denotes operator norm). In particular, if Gelander’s conjecture on the homotopy types of locally symmetric spaces holds then we get that for a symmetric space \( X \not= {\mathbb H}^3 \) there exists a \( C(X) \) such that for all finite-volume \( X \)-manifolds \( M \) we have:
\[
\log | H^q(M ; {\mathbb Z})_{\mathrm{tors}} | \le C(X) \mathrm{vol}(M).
\]

Towards analytic torsion

Let \( C^*, d_* \) be a finite complex of free finite rank \( {\mathbb Z} \)-modules, such that each \( C^q \otimes {\mathbb R} \) is endowed with an inner product with covolume \( \mathrm{vol}(C^q) = 1 \) (for example one coming from a \( {\mathbb Z} \)-basis). Define the « regulators » of the complex to be \( R^q = \mathrm{vol}(H^q(C^*)_{\mathrm{free}}) \) where \( H^q(C^*)_{\mathrm{free}} \) is viewed as a lattice in the space of harmonic cochains (orthogonal supplement of \( \mathrm{Im}(d_{q-1}) \) in \( \ker(d_q) \)). Then the metric rank formula implies that:
\[
\prod_{q=0}^d \left( \det{}'(d_q) \right)^{(-1)^q} = \prod_{q=0}^d \left( R^q / |H^q(C^*)_{\mathrm{tors}}| \right)^{(-1)^q}.
\]

We can attempt to reproduce this formula in the setting of Riemannian manifolds. Let \( M \) be a closed Riemannian manifold, and \( \mathscr L \) a local system on \( M \) associated to a representation \( \pi_1(M) \to \mathrm{SL}(L) \) where \( L \) is a finite-rank free \( {\mathbb Z} \)-module. Then for a triangulation \( K \) of \( M \) we get a cochain complex
\[
C^*(K ; \mathscr L) = \mathrm{Hom}_{\pi_1(K)} (C^*(\widetilde K), L)
\]
and cohomology groups \( H^*(K, \mathscr L) \).

There is also a corresponding de Rham complex \( C^*(M, \mathscr L_{\mathbb C}), d_*^{dR} \) with homology \( H^*(M; \mathscr L_{\mathbb C}) \) and de Rham’s theorem states that we have \( H^*(K; \mathscr L) \otimes {\mathbb C} \cong H^*(M; \mathscr L_{\mathbb C}) \).
Now suppose that \( \mathscr L_{\mathbb C} \) has an Hermitian inner product; it can be used to define a pre-Hilbert structure on the spaces of differential forms \( C^q(M; \mathscr L_{\mathbb C}) \), and in this setting we have the following theorem.

Theorem (Cheeger, Müller): Let \( \det'(d_q^{dR}) \) be the regularised determinants and \( \overline R^q \) be the covolume of \( H^q(K; \mathscr L) \) viewed as a lattice in the space of harmonic q-differential forms. Then we have:
\[
\prod_{q=0}^{\dim M} \det{}'(d_q^{dR})^{(-1)^q} = \prod_{q=0}^{\dim M} \left( \frac{\overline R^q} {| H^q(K; \mathscr L)_{\mathrm{tors}} |} \right)^{(-1)^q}.
\]

To define the regularised determinant we use the spectral zeta functions:
\[
\zeta_{d_q^{dR}}(s) = \sum_{j\ge 1} \lambda_j^{-s/2}
\]
where \( \lambda_j, j \ge 1 \) are the positive eigenvalues of the essentially self-adjoint operator \( (d_q^{dR})^* d_q^{dR} \). This converges on the half-plane \( \mathrm{Re}(s) > \dim(M)/2 \) and it extends meromorphically to \( {\mathbb C} \) to a function which does not have a pole at 0. The determinant is then defined by:
\[
\det{}'(d_q^{dR}) = \exp(-\zeta_{d_q^{dR}}'(0))
\]
(the right-hand side is formally the product of all \( \lambda_j \)).

The usual formulation of the Cheeger–Müller theorem uses the Hodge Laplacians \( \Delta_q \) rather than the differentials. Their regularised determinants are defined as above, using the spectral zeta functions \( \zeta_q \). The right-hand side of the equality in the theorem can then be rewritten as \( \exp(\tau_M(\mathscr L_{\mathbb C})) \) where:
\[
\tau_M(\mathscr L_{\mathbb C}) = \frac 1 2 \sum_{q=0}^{\dim(M)} q(-1)^q \log \det{}'(\Delta_q)
\]
is the analytic torsion of \( M \) with coefficients in \( \mathscr L_{\mathbb C} \). The Cheeger–Müller theorem can then be rewritten as:
\[
-\tau_M(\mathscr L_{\mathbb C}) = \sum_{q=0}^{\dim(M)} (-1)^q \log | H^q(K ; \mathscr L)_{\mathrm{tors}} | – \sum_{q=0}^{\dim(M)} (-1)^q \log(\overline R^q).
\]

A toy example

Let \( M = \mathbb S^1 \), \( K \) the triangulation with 1 vertex and 1 edge, and let \( \mathscr L \) be the local system associated to the representation \( \pi_1(M) = {\mathbb Z} \to \mathrm{SL}_m({\mathbb Z}) \) mapping 1 to \( A \), a semisimple matrix with \( \ker(A – 1) = \{0\} \).

The cochain complex is then given by
\[
0 \to C^0(K; \mathscr L) \cong {\mathbb Z}^m \overset{A-1}{\to} C^1(K; \mathscr L) \cong {\mathbb Z}^m \to 0
\]
(the 0-chains are functions \( c: {\mathbb Z} \to {\mathbb Z}^m \) satisfying \( c(x+1) = Ac(x) \) and the differential maps \( c \) to \( c(\cdot) – c(\cdot -1) \)). We thus see that
\[
H^*(K, \mathscr L) = 0 \oplus {\mathbb Z}^m/(A-1){\mathbb Z}^m.
\]

The de Rham complex can be computed by:
\[
0 \to C^0(M; \mathscr L_{\mathbb C}) \overset{d_0^{dR}}{\to} C^1(M; \mathscr L_{\mathbb C}) \to 0
\]
where \( C^0, C^1 \) are both identified with \( 2\pi \)-periodic functions \( {\mathbb R} \to {\mathbb R} \) and \( d_0^{dR} = d/dx + 2\pi iB\) where we have chosen \( B \in M_m({\mathbb C}) \) with \( e^{2\pi iB} = A \) (a section is a function \( f: {\mathbb R} \to {\mathbb C}^m\) satisfying \( f(x+1) = Af(x) \) a 1-form is \( f(x)dx \) where \( f \) is such a function, and the differential is \( f \mapsto df \), then we can identify this with the complex above via \( f \mapsto e^{-2\pi B\cdot}f \)). The eigenvalues of \( d_0 \) are the complex numbers \( 2\pi(n + \lambda_j), n \in {\mathbb Z} \) and it follows that:
\[
\det(d_0^{dR}) = \det(A – 1) \cdot \prod_{n \ge 1}(2\pi n)
\]
where we use the convention that \( \prod_{n \ge 1}(2\pi n) = e^{\zeta(-1)} \), where \( \zeta \) is a multiple of the Riemann zeta function. With the right normalisation we get \( \det(d_0^{dR}) = \det(A – 1) \), which is the order of \( H^1(K, \mathscr L) \).