Atiyah Conjectures and Irrational-Covolume Lattices for TDLC Groups

Definitions and mathematical statement

A totally disconnected locally compact group (TDLC group) is a Hausdorff locally compact topological group whose connected components are singletons. Such a group is compactly generated if some compact subset generates it as a group. A lattice in a locally compact group $G$ is a discrete subgroup $\Gamma\leq G$ for which $G/\Gamma$ carries a nonzero finite $G$-invariant measure. Given a Haar measure $\mu$ on $G$, the covolume $\operatorname{covol}_{\mu}(\Gamma)$ is the total mass of the compatible quotient measure, using counting measure on $\Gamma$.

The supplied record asks the following two questions.

1. Does a suitable version of the Atiyah conjecture hold for TDLC groups? The cleaned statement additionally asks whether TDLC groups can have irrational $L^2$-Betti numbers. Writing $b_n^{(2)}(G,\mu)$ for the intended $L^2$-Betti number in degree $n$, whenever it is defined, the irrationality question concerns the existence of an admissible group, measure, and degree with \[ b_n^{(2)}(G,\mu)\in{\mathbb{R}}\setminus{\mathbb{Q}}. \] Here the admissible class, the construction of these invariants, the Haar normalization, and the precise Atiyah assertion are not specified. This display therefore describes the requested phenomenon, rather than a fully determined conjecture. An Atiyah assertion might impose restrictions stronger than rationality, so the two phrasings cannot be treated as a specified logical dichotomy.
2. Does there exist a compactly generated TDLC group $G$ with a lattice $\Gamma$ of irrational covolume? This question requires a Haar normalization: replacing $\mu$ by $c\mu$, for $c>0$, gives \[ \operatorname{covol}_{c\mu}(\Gamma) =c\,\operatorname{covol}_{\mu}(\Gamma). \] Consequently, unrestricted rescaling can change the rationality of any positive finite covolume. One possible convention is to specify a compact open subgroup $K\leq G$ and normalize Haar measure by $\mu_K(K)=1$. Under that convention, the question would be whether \[ \exists\,G,K,\Gamma:\quad \begin{gathered} G\text{ is a compactly generated TDLC group},\\ K\leq G\text{ is compact and open},\qquad \Gamma\leq G\text{ is a lattice},\\ \operatorname{covol}_{\mu_K}(\Gamma)\in{\mathbb{R}}\setminus{\mathbb{Q}}. \end{gathered} \] This is an explicitly conditional formulation: the supplied record does not state that normalization or otherwise resolve the scale dependence.

The first question does not explicitly assume compact generation; that hypothesis is supplied only for the lattice question. No second countability, unimodularity, finiteness condition for the intended $L^2$-invariants, or restriction to uniform lattices is explicitly stated. Any hypotheses required by the intended theory must be recovered from the source before a definitive formulation is accepted.

The original extract continues into a section headed Yuri Santos Rego, Questions 8 and 9, about virtually RFRS Coxeter groups and profinite completions. Those appear to be adjacent material rather than additional clauses of the two TDLC questions; their attachment and attribution remain unverified.

Short English statement

The record asks for an Atiyah-type statement for totally disconnected locally compact groups, including a question about irrational $L^2$-Betti numbers, and asks whether a compactly generated such group can have a lattice of irrational covolume. A precise formulation needs the intended $L^2$-theory, hypotheses, and Haar normalization. The supplied open status and verification claims are curation labels requiring human review.

Sources

• Local UnsolvedMath registry, supplied complete record with permanent ID $30005707$, exact source problem code catalogue entry, title "Atiyah Conjectures and Irrational-Covolume Lattices for TDLC Groups." The record has no named proposer and supplies proposed year 2023.
• Supplied source citation: Mini-Workshop: Homological aspects of TDLC-groups (2024), Oberwolfach Reports, pp. 3103--3138, DOI 10.4171/owr/2023/55. https://doi.org/10.4171/owr/2023/55. The citation's displayed year 2024 differs from the proposed year and the year embedded in the DOI; the supplied bibliographic details are retained without reconciliation.
• Supplied literature source, labeled "Oberwolfach report": https://ems.press/content/serial-article-files/48166.
• Supplied literature source, labeled "Hughes--Moller--Varghese": https://arxiv.org/abs/2505.08701. No title or precise result is supplied for this reference.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that the Atiyah statement is known for important discrete and Coxeter subclasses, while the TDLC version and the irrational-covolume lattice question remain unresolved. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The record labels its statement corrected_verified and reports checking it against the official Oberwolfach report. These are supplied source-verification claims; no independent source visit or verification is asserted here.