[Paper Review] Approximating L^2 invariants of amenable covering spaces: A heat kernel approach

Józef Dodziuk, Varghese Mathai|ArXiv.org|Sep 6, 1996

Geometric and Algebraic Topology9 references20 citations

TL;DR

This paper establishes that $L^2$ Betti numbers of amenable covering spaces can be approximated by the normalized Betti numbers of a regular exhaustion of the universal cover, using a heat kernel approach based on the principle of not feeling the boundary. The key result is that under suitable geometric conditions, the limit of averaged Betti numbers equals the $L^2$ Betti number, with equality holding in dimension ≤4 under mild cohomological assumptions.

ABSTRACT

In this paper, we prove that the L^2 Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, under some hypotheses. We also prove that some L^2 spectral invariants can be approximated by the corresponding average spectral invariants of a regular exhaustion. The main tool which is used is a generalisation of the "principle of not feeling the boundary" (due to M. Kac), for heat kernels associated to boundary value problems.

Motivation & Objective

• To establish approximation theorems for $L^2$ Betti numbers of amenable Galois covering spaces using heat kernel techniques.
• To investigate whether the normalized Betti numbers of a regular exhaustion converge to the $L^2$ Betti numbers in the limit.
• To extend the principle of not feeling the boundary to boundary value problems on Riemannian manifolds with controlled geometry.
• To prove that the integrated density of states for Laplacians on $L^2$-forms is independent of boundary conditions (relative or absolute) on amenable covers.
• To provide evidence for a conjecture that normalized Betti numbers converge to $L^2$ Betti numbers for all amenable covers, with proof in dimensions ≤4 under cohomological finiteness.

Proposed method

• Uses a generalization of the 'principle of not feeling the boundary'—originally due to M. Kac—for heat kernels on manifolds with boundary.
• Applies the heat kernel method to Laplacians on $L^2$-forms with relative and absolute boundary conditions on a regular exhaustion $\{D_k\}$ of the universal cover $\widehat{M}$.
• Imposes geometric control on the exhaustion: vanishing relative isoperimetric ratio, uniformly bounded second fundamental forms, and uniform collaring of boundaries.
• Relies on spectral density functions $N_j^{(k)}(\lambda)$ and their convergence to the $L^2$ spectral density $N_{j,\Gamma}(\lambda)$ as $k \to \infty$.
• Uses zeta function regularization to study convergence of spectral invariants, with $\zeta(s, \Delta_p^{(k)} + \lambda)$ converging to $\zeta(s, \Delta_p + \lambda)$ in the half-plane $\Re(s) > n/2$.
• Establishes that the integrated density of states is independent of boundary conditions via the relation $F_p(\lambda) = G_{p+1}(\lambda)$ for amenable groups.

Experimental results

Research questions

• RQ1 Can $L^2$ Betti numbers of amenable covering spaces be approximated by normalized Betti numbers of a regular exhaustion?
• RQ2 Does the limit of normalized Betti numbers (absolute or relative) equal the $L^2$ Betti number in the amenable case?
• RQ3 Is the integrated density of states for the Laplacian on $L^2$-forms independent of boundary conditions on amenable covers?
• RQ4 Under what conditions does the zeta function of the Laplacian on $D_k$ converge to the $L^2$ zeta function on $\widehat{M}$?
• RQ5 What is the relationship between the Novikov-Shubin invariants and the spectral density growth rate in the amenable setting?

Key findings

• For amenable Galois covers, the limit superior of normalized Betti numbers satisfies $\limsup_{k\to\infty} \frac{\mathrm{vol}(M)}{\mathrm{vol}(D_k)} b^j(D_k, \partial D_k) \leq b_{(2)}^j(\widehat{M}, \Gamma)$, with equality when $b_{(2)}^j = 0$.
• In dimension 2, the conjecture on convergence of normalized Betti numbers to $L^2$ Betti numbers holds unconditionally.
• For dimensions 3 and 4, the conjecture holds under the assumption that $\dim H^1(\widehat{M}) < \infty$.
• The integrated density of states for $\Delta_j$ is independent of boundary conditions (relative or absolute), as shown by $F_p(\lambda) = G_{p+1}(\lambda)$.
• The zeta function $\zeta(s, \Delta_p^{(k)} + \lambda)$ converges uniformly on compact subsets of $\Re(s) > n/2$ to $\zeta(s, \Delta_p + \lambda)$ as $k \to \infty$.
• If the spectral density satisfies $\frac{\mathrm{vol}(M)}{\mathrm{vol}(D_k)}[N_j^{(k)}(\lambda) - N_j^{(k)}(0)] \leq C\lambda^{\beta_j}$, then the $j$-th Novikov-Shubin invariant satisfies $\alpha_j(\widehat{M}) \geq \beta_j$, and the main conjecture holds.