Automaton Kernel Series and $\ell^2$-Betti Values

Definitions and mathematical statement

Let $n$ and $C$ be the integer parameters in the supplied statement. The matrix alphabet is \[ \Sigma_{n,C}=\{B\in\operatorname{Mat}_{n\times n}({\mathbb{Z}}): B_{ij}\in\{-C,-C+1,\ldots,C\}\}. \] The intended parameter restrictions appear to be $n\geq1$ and $C\geq0$; these restrictions are not explicitly stated in the record. Let $A$ be the supplied automaton over this alphabet, and write $L(A)$ for its accepted language. The source does not specify the automaton model.

The source describes a matrix $M$ of dimension $d(M)\geq n$ as recognized when its entries vanish outside the $n$-neighborhood of the diagonal and $A$ accepts its word of consecutive $n\times n$ diagonal minors. Here the letters must be matrix blocks, rather than determinants, to belong to $\Sigma_{n,C}$. The neighborhood and block-reading conventions require clarification.

For precision, one possible interpretation can be expressed conditionally as follows. If the matrices are square integer matrices and consecutive blocks mean overlapping principal blocks, define \[ W_n(M)=\left(M[\{j,\ldots,j+n-1\},\{j,\ldots,j+n-1\}]\right)_{j=1}^{d-n+1} \quad(M\in\operatorname{Mat}_{d\times d}({\mathbb{Z}})). \] For a specified bandwidth $b$, the corresponding recognized set would be \[ \mathcal R_{A,d}^{(b)}= \left\{M\in\operatorname{Mat}_{d\times d}({\mathbb{Z}}): M_{ij}=0\text{ if }|i-j|>b,\quad W_n(M)\in L(A)\right\}. \] The readings $b=n$ and $b=n-1$ are different; the supplied record does not resolve which is intended, nor confirm the overlapping-block convention. In particular, overlapping blocks of size $n$ do not inspect entries with $|i-j|=n$, so the alphabet bound alone does not bound those entries under the reading $b=n$. This conditional formulation is not a choice of the intended convention.

Once the recognized sets $\mathcal R_{A,d}$ have been specified, count each recognized matrix once and set \[ a_d(A)=\sum_{M\in\mathcal R_{A,d}}\dim_{{\mathbb{Q}}}\ker(M:{\mathbb{Q}}^d\longrightarrow{\mathbb{Q}}^d), \qquad F_A(x)=\sum_{d\geq n}a_d(A)x^d, \] provided every coefficient is finite. For integer matrices, the nullity is unchanged by extending scalars to ${\mathbb{R}}$ or ${\mathbb{C}}$. The integer-matrix setting and characteristic-zero interpretation are explicit conditional conventions here, since the source does not separately specify the ambient matrix domain or kernel field.

Automaton-series question. With the intended recognition conventions fixed, determine the possible values of $F_A(q)$ at rational $q$. In particular, do there exist permitted parameters, an automaton $A$, and $q\in{\mathbb{Q}}$ such that \[ F_A(q)\in\overline{{\mathbb{Q}}}\cap{\mathbb{R}}\setminus{\mathbb{Q}}? \] Here $\overline{{\mathbb{Q}}}$ denotes the algebraic numbers. Under the ordinary convergent-power-series interpretation, evaluation requires \[ \sum_{d\geq n}a_d(A)|q|^d<\infty. \] The supplied question gives no convergence condition or alternative evaluation rule, so this evaluation convention also requires confirmation.

The original extracted statement additionally asks \[ C(G)\not\subseteq{\mathbb{Q}} \quad\Longleftrightarrow\quad {\mathbb{Z}}/p\wr K\subseteq G \text{ for some prime }p\text{ and infinite finitely generated group }K, \] and asks for $C({\mathbb{Z}}/2\wr{\mathbb{Z}})$. These are retained as source questions: the record does not define $C(G)$, specify the class of groups $G$, or specify the wreath-product convention. Consequently, they cannot be made into fully defined mathematical targets from the supplied record alone.

Short English statement

The main question asks which numbers arise by evaluating a generating series that counts kernel dimensions of matrices recognized by an automaton, especially whether an irrational algebraic value can occur at a rational argument. The recognition conventions, matrix domain, and evaluation domain need clarification. The original record also contains group-theoretic questions involving an undefined invariant $C(G)$. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record: data/unsolvedmath/problems.json, permanent ID $30001585$, title "Automaton Kernel Series and $\ell^2$-Betti Values," exact source code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is supplied; the proposed year is recorded as 2010.
• Supplied authoritative source 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 supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions.
• Supplied literature reference: Grabowski, Atiyah problem for lamplighter groups, https://arxiv.org/abs/1009.0229.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the record "partially solved" and reports that explicit transcendental-kernel constructions settle the existence of non-rational values for the lamplighter group, while the broader subgroup characterization and automata-series questions remain open. This is supplied dated curation, not an independently verified literature assessment; its time-sensitive status and cited claims require independent checking.
• The supplied registry labels the statement corrected_verified and reports that its mathematical content was checked against the official Oberwolfach report, with notation, context, and question form normalized. These are supplied verification claims; no source was independently visited or checked in preparing this draft.