Arithmetic Cofinitary Representation

• Arithmetic cofinitary representation embeds abstract groups into S∞ using first-order arithmetic formulas, ensuring every nonidentity element fixes only finitely many points.
• The construction partitions ω into finite intervals and applies left-regular actions to achieve a Σ⁰₂-definable, cofinitary embedding of large abelian groups.
• These representations link permutation group theory, logic, and infinite combinatorics, offering a framework for studying definability and maximal cofinitary groups.

An arithmetic cofinitary representation is a highly structured embedding of an abstract group into the infinite symmetric group $S_\infty$ such that both the group action and generating set are definable by first-order arithmetic formulas, and the image acts cofinitarily—meaning every non-identity element has only finitely many fixed points. These representations play a fundamental role in connecting permutation group theory, logic (especially definability and the arithmetical hierarchy), and infinite combinatorics (Schembecker, 30 Dec 2025, Schrittesser, 2021).

1. Foundational Concepts

A cofinitary group is a subgroup $G\leq S_\infty$ such that every $g\in G\setminus\{e\}$ has finitely many fixed points: $\left|\{n\in\omega: g(n)=n\}\right|<\infty$. A group is maximal cofinitary if it is not properly contained in a larger cofinitary subgroup.

An arithmetic cofinitary representation of an abstract group $H$ is an embedding $H\hookrightarrow S_\infty$ such that the set of generators $\Gamma\subseteq\omega^\omega$ and the action $(g,n)\mapsto g(n)$ are arithmetical; that is, they are definable by first-order formulas in the structure $(\omega,+,\times)$. For technical purposes, typically one ensures $\Gamma$ is $\Sigma^0_2$-definable in the arithmetical hierarchy (Schembecker, 30 Dec 2025).

2. Main Existence Theorems

The pioneering result for arithmetic cofinitary representations is the unconditional construction that the large abelian group $H=\bigoplus_{f\in{}^\omega 2} \mathbb{Z}_2$ (the direct sum of continuum many copies of $\mathbb{Z}_2$) admits such a representation. Explicitly, there exists a $\Sigma^0_2$ arithmetical set $\Gamma\subseteq S_\infty$ such that the group it generates is isomorphic to $H$ and acts cofinitarily on $\omega$ (Schembecker, 30 Dec 2025).

The construction refines earlier consistency arguments (such as Kastermans's for $\bigoplus_{\aleph_1} \mathbb{Z}_2$ under additional set-theoretic hypotheses) to a theorem provable in ZFC. Furthermore, it bypasses the need for forcing or extra axioms and achieves definitions at low arithmetical complexity (Schembecker, 30 Dec 2025, Schrittesser, 2021).

3. Arithmetic Construction of the Representation

The essential construction proceeds in the following steps (Schembecker, 30 Dec 2025):

1. Presentation of $H$: Write $H=\bigoplus_{f\in2^\omega} \langle h_f \rangle$ with each $h_f^2=e$ and each pair $h_f,h_g$ commuting.
2. Partition of $\omega$: Partition $\omega$ into disjoint finite intervals $I_n$ such that $|I_n|=2^n$ and $\omega=\bigsqcup_{n<\omega} I_n$.
3. Local Actions: For each $n$, let $H_n=\bigoplus_{s\in2^n} \langle h_s\rangle$, and identify $I_n$ with $H_n$. By Cayley’s theorem, the left-regular action of $H_n$ on itself is free.
4. Global Action: Define the action of each generator $h_f$ on $k\in I_n$ by $h_f\cdot k = h_{f\restriction n}(k)$, where $h_{f\restriction n} \in H_n$ acts by left multiplication. The construction ensures that relations $h_f^2=e$ and $h_f h_g = h_g h_f$ hold, and that any nontrivial element acts without fixed points outside a finite set.
5. Cofinitarity Verification: For $h=\sum_{f\in F_0} h_f \neq e$, there exists $N$ such that the restriction of $h$ to each $I_n$ with $n>N$ is nontrivial and thus fixes no points, so $h$ has only finitely many fixed points in $\bigcup_{m<N} I_m$.

4. Definability and Arithmetical Complexity

The representation is arithmetic because the generating set

$$\Gamma = \{\,g \in S_\infty : \forall n\,\exists s\in 2^n\,\forall k\in I_n\, [g(k)=h_s(k)]\,\}$$

is $\Sigma^0_2$-definable in $(\omega,+,\times)$. Each step—verifying agreement with $h_s$ on $I_n$—uses only bounded quantification. Thus, the set $\Gamma$ and the action map are arithmetical, and the group they generate is arithmetically presented (Schembecker, 30 Dec 2025).

This definability is significant: it positions arithmetic cofinitary representations low in the descriptive complexity hierarchy, consistently below $K_\sigma$ subgroups, and well below analytic ($\Sigma^1_1$) or general Borel complexity (Schrittesser, 2021).

5. Relation to Maximal Cofinitary Groups and Further Directions

While the abelian group $H$ constructed above cannot be maximal cofinitary (by Blass’s results), product-with-finite-group modifications and further surgery techniques enable the construction of arithmetic maximal cofinitary groups of isomorphism type $(*_{\mathfrak{c}} \mathbb{Z}) \times F$ for any finite $F$ (Schembecker, 30 Dec 2025). In particular, there exist arithmetic maximal cofinitary groups whose defining property is captured by an arithmetical formula with only a finite number of quantifiers over $\omega$ (Schrittesser, 2021).

A central open question concerns the minimal descriptive complexity of a maximal cofinitary group. Current constructions yield $\Sigma^0_2$-definable (finite-level Borel) maximal cofinitary groups. It remains unresolved whether truly closed ($\Pi^0_1$) or effectively closed maximal cofinitary groups exist (Schrittesser, 2021). Kastermans established that no $K_\sigma$ subgroup of $S_\infty$ can be maximal cofinitary, suggesting intrinsic limitations tied to definability.

6. Broader Mathematical Context and Implications

Arithmetic cofinitary representations intertwine group theory, recursion theory, and descriptive set theory. They provide highly uniform witnesses for the existence of large cofinitary permutation groups and demonstrate the reach of first-order arithmetic in encoding group-theoretic structures on infinite sets.

A plausible implication is the utility of these constructions as templates in related combinatorial and logic-theoretic contexts where definability, hierarchy placement, and effective coding are essential. Their role in answering previously open questions about isomorphism types and definability for maximal cofinitary groups highlights their foundational significance within infinite group theory (Schembecker, 30 Dec 2025, Schrittesser, 2021).