Cayley Correspondence: Algebraic and Combinatorial Limits

• Cayley correspondence is a guiding principle connecting group theory, combinatorics, and geometry by encoding group actions and symmetries through Cayley graphs and diagrams.
• It reveals that local convergence of finite unlabeled graphs can approximate a Cayley graph while failing to capture detailed labeled structures essential for complete group presentations.
• Case studies show that subtle differences in labeling and orientation can lead to loss of global algebraic information, impacting testability and the study of sofic groups.

A Cayley correspondence is a structural relationship linking algebraic, geometric, or combinatorial objects through the lens of group actions, groupoids, or related symmetries, where the associated Cayley graphs or diagrams offer a unifying framework. Depending on context, the correspondence characterizes how properties of groups, graphs, polytopes, algebraic varieties, or moduli spaces are encoded or reflected in each other. The concept encompasses results showing that when objects share “Cayley” symmetries, their descriptive invariants—be they combinatorial, topological, or geometric—are often tightly interconnected, but can also exhibit significant divergences depending on the nature of additional structure (such as edge labeling or orientation). Cayley correspondence is not a single theorem but a guiding principle appearing in group theory, combinatorics, Higgs bundle theory, and the theory of operator algebras.

1. Algebraic and Combinatorial Foundations

The Cayley correspondence originates in the theory of Cayley graphs and Cayley diagrams. Given a finitely generated group $\Gamma$ and a generating set $S$, the Cayley graph $G = \operatorname{Cay}(\Gamma, S)$ is the undirected graph with vertex set $\Gamma$ and edge set

$E = \{ \{g, gs\} : g \in \Gamma,\, s \in S \}.$

This construction “visualizes” the group structure as a metric graph. The richer structure of a Cayley diagram encodes, via labeling and edge orientation, the specific generator $s \in S$ used at each step, which amounts to a directed, labeled multigraph maintaining the full data of the group presentation.

The paper (Timar, 2011) emphasizes that weak convergence (Benjamini–Schramm convergence) of finite graph sequences may yield a Cayley graph as limit, but the corresponding convergence on diagrams (i.e., the “labeled, oriented” structure) is strictly finer. The metric for rooted isomorphism classes of graphs/diagrams is defined via maximal isomorphic neighborhoods of bounded radius, with distance $2^{-r}$ for $r$ the maximal such radius. Weak convergence is thus local convergence, and the probability measure on finite-radius rooted neighborhoods encodes the limiting object.

A critical distinction arises: sequences of finite graphs may converge to a given Cayley graph, but no sequence of labeled diagrams (with consistent local labels/orientations) may converge to the associated Cayley diagram. Approximability in the unlabeled case does not imply approximability in the labeled (diagram) case. The theoretical implications for soficity and the recoverability of group structure from combinatorial limits are profound.

2. Limiting Behavior and Counterexamples

Key constructions in (Timar, 2011) demonstrate the separation between unlabeled and labeled convergence:

• There exists a Cayley diagram $G$ for which a sequence of finite graphs $G_n$ (e.g., of the form $H_n \times C_4$ with $H_n$ 3-regular graphs of large girth and bounded independence ratio) converges weakly to the underlying Cayley graph of $G$, but no labeling on $G_n$ induces convergence to the diagram $G$.
• This is achieved by constructing product graphs $T \times C_4$ (with $T$ a 3-regular tree and $C_4$ the cycle), whose fibers are decorated with distinct local diagrams and labelings preclude a consistent diagrammatic limit.
• Furthermore, sequences $G_n$ and $K_n$ may both converge to a Cayley graph $G$, but only $K_n$ contains a marked “subdiagram” (such as a Hamiltonian cycle) converging to a bi-infinite path in $G$. In $G_n$, no such subgraph sequence converges.

3. Implications for Testability and Soficity

These constructions have direct consequences for testability in property testing of sparse graphs. For example:

• The presence of a Hamiltonian cycle is not a “testable” graph property in Benjamini–Schramm convergence: local neighborhoods cannot distinguish between the presence or absence of such a global structure in the limit.
• The results reveal a key separation: approximability of the unlabeled Cayley graph does not imply approximability of the labeled diagram, nor does it guarantee that the associated group is sofic. Whether Cayley graph approximability by finite graphs implies soficity remains an open question of central importance in geometric group theory.

This leads to a broader application: the necessity of careful analysis of which combinatorial/geometric properties are “local” (and hence captured by limits of finite graphs) and which are fundamentally global (and lost in the passage to local weak limits).

4. Mathematical Rigor and Proof Outline

Mathematical details in (Timar, 2011) elaborate:

• The rooted graph distance and weak convergence are formalized via

$d((G,o),(H,o')) = 2^{-r}$

for maximal $r$ such that the rooted $r$-balls are isomorphic.

• For a sequence $\{G_n\}$ of finite graphs (chosen with uniform random root), weak convergence means

$$\lim_{n\to\infty} \int f \, d\mu_{G_n} = \int f \, d\mu$$

for all bounded continuous $f$; if $\mu$ is Dirac on a transitive graph, this is classical graph approximability.

• In Theorem 3, the direct product $T \times C_4$ is used, with labelings on the $C_4$ factor (using two colors $c$ and $d$) not allowing diagrammatic convergence due to independent set properties arising from fiber orientation. In Theorem 5, the inability to approximate a bi-infinite Hamiltonian cycle subgraph via $G_n$ is proved with careful combinatorial and independence arguments.

5. Subtlety of Cayley Correspondence

The main conceptual outcome is that the Cayley correspondence—between graph theoretical, combinatorial, and group-theoretic limits—does not align as simply as one may expect. While Cayley graphs are naturally approximated by finite graphs (through local convergence), the Cayley diagram—encoding the entire group presentation—admits a strictly stronger notion, and limits here can “forget” non-local group structure unless labelings are compatible.

As such, the preservation of algebraic data (generators and their relations) within local limits is highly sensitive to labeling and orientation, and not all global combinatorial/data-theoretic features survive the limiting operation. This has probabilistic ramifications for random graph models and for the realization of graph properties that rely on these global features.

6. Broader Context and Open Directions

The findings raise important avenues for further research:

• Question 2 in (Timar, 2011): Does approximability of a Cayley graph by finite graphs (as unlabeled objects) imply soficity of the underlying group? This remains open and is closely related to the paper of sofic groups in group theory.
• The separation between local convergence and the preservation of marked (e.g., colored) substructures is directly relevant to parameter testing in sparse graphs and geometric group theory.
• Understanding the alignment (or misalignment) of graph and diagram approximability impacts combinatorial optimization and algorithms in group-based structure identification.

An explicit recognition is required: the passage from algebraic structure (groups and presentations via Cayley diagrams) to combinatorial structure (graphs and their local subgraphs) is not, in general, lossless under local convergence, and delicate counterexamples exist even for natural group and graph constructions.

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The detailed analysis in (Timar, 2011) demonstrates that Cayley correspondence is a nuanced and context-dependent phenomenon, highlighting the intricate interplay between algebraic, combinatorial, and topological properties in the paper of infinite group presentations, their finite approximations, and the structure of their associated graphs and diagrams.