$L^2$-Invariants of Finite-Volume Locally Symmetric Spaces

Definitions and mathematical statement

The source concerns finite-volume locally symmetric spaces and asks whether their $L^2$-Betti numbers and $L^2$-torsion come entirely from the "complete interior." It also requests an extension of existing calculations first to rank-one and then to higher-rank locally symmetric spaces. The latter extension request is restored in the cleaned statement and retained separately from the interior-contribution question.

In a conventional manifold setup one writes $M=\Gamma\backslash G/K$, with $G/K$ a symmetric space and $\Gamma$ a torsion-free lattice, and uses its locally symmetric metric of finite volume. The record does not itself specify the groups, metric normalization, torsion/orbifold convention or compact and noncompact cases covered. The ranks in the extension request also need definition: real rank of the symmetric space, commonly the dimension of a maximal flat, is not interchangeable with an arithmetic ${\mathbb{Q}}$-rank controlling cusps.

There are distinct $L^2$-cohomology conventions to resolve. On $M$, reduced $L^2$ de Rham cohomology is a Hilbert-space cohomology whose dimension can be compared with the kernel of a suitably realized Hodge Laplacian. For a regular cover $\widetilde M\to M$ with deck group $\Gamma$, the group $L^2$-Betti numbers instead use $\dim_\Gamma\ker\Delta_i$ in the corresponding setup. The supplied record does not say which version and cover are intended. $L^2$-torsion further requires its analytic or combinatorial definition, trace/normalization, and the relevant determinant, spectral or acyclicity hypotheses (or a prescribed modification). These are not supplied and cannot be presumed in every finite-volume example.

Most importantly, "complete interior" is not defined as a comparison space, region or spectral object. A complete noncompact manifold, a compact truncated core with boundary conditions, and a completed interior of a compactification are not automatically the same comparison problem. To state that the invariants come entirely from it, one needs the proposed equality or decomposition formula and a precise meaning of contributions from the ends. No boundary condition, vanishing of an end invariant, or equality with invariants of a compact core is guessed.

The literature assessment checked 2026-08-21 reports calculations in compact and selected noncompact cases but says they do not settle the broad ends/interior question. The verification field reports restoring both rank-extension tasks and the interior conjecture from the report. These and the open label are dated source metadata; no universal formula or independently verified current status is asserted.

Short English statement

Determine whether the stated $L^2$-invariants are entirely accounted for by the source's complete interior, and extend the calculations through rank one to higher rank. A precise comparison needs the meaning of interior and ends, the chosen $L^2$ theory and the torsion hypotheses. These definitions are absent, so no compact-core or end-vanishing statement is substituted.

Sources

• Local UnsolvedMath registry, permanent ID 30000431, source code catalogue entry, "$L^2$-Invariants of Finite-Volume Locally Symmetric Spaces," including the original interior question and the cleaned rank-one/higher-rank extension request.
• Miniworkshop: L2-Spectral Invariants and the Integrated Density of States (2006), Oberwolfach Reports, pp. 511--552, DOI 10.4171/owr/2006/09: https://doi.org/10.4171/owr/2006/09.
• Supplied reference "Olbrich L2 invariants": https://arxiv.org/abs/math/0009039.
• Literature triage checked 2026-08-21; registry record updated 2026-09-16. No external source has been visited or independently checked here.