What the critical group sees of a harmonic action

Definitions and mathematical statement

Item 41 (Lionel Levine). Study the extra automorphism-group structure on the critical group of a graph carrying a harmonic action. The source is a research suggestion, not a fixed conjecture or requested classification. The registry says open/OPEN-TRIAGE, while the aid labels the item context-only/not-a-problem.

For a finite connected loopless multigraph $X$, let $\operatorname{Div}^0(X)$ be degree-zero integral vertex divisors and \[ \operatorname{Jac}(X)=\operatorname{Div}^0(X)/\operatorname{im}\Delta \] its critical group. Every graph automorphism acts on vertex divisors and commutes with the Laplacian, inducing $\operatorname{Aut}(X)\to\operatorname{Aut}(\operatorname{Jac}(X))$. A specified group action therefore gives a finite group module. A usual harmonic-action convention, without edge inversions, requires vertex stabilizers to act freely on incident half-edges; the original does not supply its convention.

Reported kernel-displacement theorem. Suppose $X$ is bridgeless with $n$ vertices. Let \[ K=\ker(\operatorname{Aut}(X)\to\operatorname{Aut}(\operatorname{Jac}(X))),\qquad P=\ker(\operatorname{Aut}(X)\to\operatorname{Sym}(V(X))). \] Vertex-fixing edge permutations lie in $K$. The aid proves that \[ d(\sigma)=[\sigma(v)-v]\quad(\sigma\in K) \] is independent of $v$, is a homomorphism, and induces the canonical injection \[ K/P\hookrightarrow\operatorname{Jac}(X)[n]. \] Here $[n]$ denotes elements annihilated by $n$. For simple bridgeless graphs $P$ is trivial, so the automorphisms invisible on the critical group form an abelian group of exponent dividing $n$. The bridge-free hypothesis is material to identifying the zero-displacement kernel with $P$.

The aid combines this with a 2025 semiregular faithfulness theorem of Estelyi--Karabas--Mednykh--Nedela to conclude that the full automorphism group of every simple connected $3$-edge-connected graph acts faithfully on its critical group. Here $3$-edge-connected means no cut of fewer than three edges disconnects the graph. The exact external theorem hypotheses and citation are absent.

Reported multigraph contrast. For any finite group $G$, take the banana multigraph with two vertices and $|G|$ parallel edges labeled by $G$. Let $G$ fix the vertices and act regularly on edge labels. This graph action is faithful and harmonic in the declared convention, while the induced action on \[ \operatorname{Jac}(X)\simeq{\mathbb{Z}}/|G|{\mathbb{Z}} \] is trivial because all vertex divisors are fixed. Thus a faithful harmonic graph action need not be detectable on the critical group when parallel-edge permutations are allowed. This contrasts with the simple $3$-edge-connected conclusion and does not classify all possible modules or settle a formal conjecture absent from the original suggestion.

Short English statement

The source suggests studying a graph's critical group together with a harmonic automorphism action. The aid gives a canonical bound on the invisible automorphisms of bridgeless graphs and reports faithfulness for simple $3$-edge-connected graphs. Parallel-edge banana examples show that arbitrary faithful harmonic graph actions can still act trivially on the critical group.

Sources

• Local UnsolvedMath registry, permanent ID $20001006$, catalogue entry; canonical aim-combinatorics-notes.json, notes[130].
• AIM workshop "Generalizations of chip-firing and the critical group," PDF item 41, Lionel Levine: https://aimath.org/pastworkshops/chipfiringproblems.pdf.
• Estelyi--Karabas--Mednykh--Nedela, 2025 semiregular faithfulness theorem, as named in the supplied summary; no full title, identifier or theorem number supplied.
• Research attempt 1, context_only, run status not_a_problem, internal rigor proved, low novelty confidence. Explicitly machine-generated, not peer reviewed, and to be independently verified before citation. Original literature field empty.