The $\ell^2$-Betti Spectrum of the Lamplighter Group

Definitions and mathematical statement

Let \[ G=({\mathbb{Z}}/2{\mathbb{Z}})\wr{\mathbb{Z}} =\left(\bigoplus_{j\in{\mathbb{Z}}}{\mathbb{Z}}/2{\mathbb{Z}}\right)\rtimes{\mathbb{Z}} \] be the lamplighter group, where the direct sum consists of finitely supported functions on {$\mathbb{Z}$} and the generator of {$\mathbb{Z}$} acts by shifting their indices.

The supplied problem asks to determine \[ C(G), \] described as the set of $\ell^2$-Betti numbers arising from $G$. The supplied record does not define which constructions or coefficient ring are allowed in forming this set; consequently, it does not uniquely specify the spectrum to be determined.

For an explicit possible formulation, fix a unital subring $K\subseteq{\mathbb{C}}$. For positive integers $m,n$ and a matrix $A\in M_{m,n}(KG)$, let \[ T_A:\ell^2(G)^n\longrightarrow\ell^2(G)^m \] be the bounded operator given by left convolution. Let $P_A$ be the orthogonal projection onto $\ker T_A$, and define its von Neumann dimension by \[ \dim_G\ker T_A =\sum_{i=1}^{n}\langle P_A\delta_{e,i},\delta_{e,i}\rangle, \] where $e$ is the identity of $G$, and $\delta_{e,i}$ is the vector supported at $e$ in the $i$-th summand. This gives the precisely defined candidate spectrum \[ C_K(G)=\left\{\dim_G\ker T_A: m,n\geq1,\ A\in M_{m,n}(KG)\right\}. \] If the source intends the rational group-ring kernel convention, its question becomes: determine $C_{{\mathbb{Q}}}(G)$ exactly. This identification is conditional: the supplied record does not establish that $C(G)=C_{{\mathbb{Q}}}(G)$, nor does it specify alternative matrix, coefficient, or topological conventions. The reference to an associated regular-language classification in the background gives no definition of that classification and therefore supplies no additional precise mathematical target.

Short English statement

Determine exactly which $\ell^2$-Betti values arise from the lamplighter group $({\mathbb{Z}}/2{\mathbb{Z}})\wr{\mathbb{Z}}$. The intended definition of the spectrum $C(G)$, including its coefficient ring and allowed constructions, requires source clarification. The rational group-ring kernel formulation above is conditional. The supplied partially solved label and literature assessment are dated curation, not independently verified conclusions.

Sources

• Local UnsolvedMath registry, supplied record with permanent ID $30001586$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2010.
• Supplied report citation: Actions and Invariants of Residually Finite Groups: Asymptotic Methods (2011), Oberwolfach Reports, pp. 2335--2391, DOI 10.4171/owr/2010/40. Original OWR report: https://doi.org/10.4171/owr/2010/40. The record describes these reports as documenting workshops and their open-problem discussions.
• Supplied literature reference: Grabowski, Atiyah problem for lamplighter groups: https://arxiv.org/abs/1009.0229.
• Supplied literature reference: Austin, rational group-ring elements: https://arxiv.org/abs/0909.2360.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports transcendental $\ell^2$-Betti values for the lamplighter group, while describing the complete set $C$ and an associated regular-language classification as undetermined. This is supplied dated curation derived from the OWR collection, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports comparison with the official Oberwolfach report, with notation, context, and question form normalized. This is a supplied verification claim; the present draft does not claim to have consulted the report or the linked literature.