Cone-stack moduli and stabilizer criteria for harmonic metric-graph quotients

Definitions and mathematical statement

Jordan Ellenberg asks for a Deligne--Mumford stack on graphs and, after Matt Baker's comment about a metric-graph paper, asks what harmonic group actions on metric graphs should mean. The source does not specify the category of graphs, families, base topology, stability, or sense of Deligne--Mumford analogue. Baker's paper is not identified.

For the aid's stable moduli interpretation, fix stable weighted marked graph types, with vertex weights $h(v)\ge0$, prescribed total genus, and stability $2h(v)-2+\operatorname{val}(v)+\#\text{marks at }v>0$. The reported tropical cone-stack cells are \[ [{\mathbb{R}}_{>0}^{E(G)}/\operatorname{Aut}(G)], \] where automorphisms preserve the weights and markings. Cells are glued using edge contractions and appropriate contracted vertex data. This is a cone-stack/tropical analogue with finite combinatorial stabilizers, not an unspecified algebraic Deligne--Mumford stack over schemes. The source's unrestricted graph-stack question is not answered merely by listing the cells without its intended category.

For the metric quotient, let a finite group $K$ act isometrically on a connected finite metric graph $\Gamma$ having at least one edge. Subdivide equivariantly to resolve inversions. For an edge $e$, let $K_e$ denote its generic point stabilizer, now fixing the edge pointwise; for a point $x$, let $K_x$ be its stabilizer. A finite metric-graph map is harmonic at $x$ if the sum of outgoing integer slopes above a chosen target tangent direction is independent of that target direction; this common sum is its local degree.

The aid reports that giving the orbit edge $\bar e$ the weighted length \[ \ell(\bar e)=|K_e|\ell(e) \] makes $\Gamma\to\Gamma/K$ harmonic, with edge slope $|K_e|$, local degree $|K_x|$, and global degree $|K|$. By contrast, the ordinary path-metric quotient gives slope one on each resolved edge. Its reported criterion is \[ \Gamma\longrightarrow\Gamma/K\text{ with the unscaled path metric is harmonic} \quad\Longleftrightarrow\quad |K_e|=s\text{ is constant over all edges}. \] When this holds, its local degree is $|K_x|/s$ and global degree is $|K|/s$. The aid says the criterion is subdivision invariant. The distinction is between two quotient metrics on the same orbit combinatorics, not a statement that arbitrary isometric actions fail to admit a harmonic weighted quotient.

The reported obstruction is a $C_2$-action on a Y-shaped graph fixing one arm and exchanging two equal-length arms. At the center the unscaled map has one preimage direction over the fixed arm and two over the exchanged-arm orbit, so the directional sums disagree. The generic stabilizers have orders two and one respectively. The weighted metric compensates this difference. The aid calls this a smallest obstruction, but does not state a graph category or minimality measure for that additional claim. These constructions remain supplied partial results requiring verification, not a complete treatment of all graph stacks or harmonic actions.

Short English statement

The source asks about graph moduli stacks and harmonic group actions. The supplied aid describes a stable tropical cone-stack analogue and distinguishes the always available stabilizer-weighted quotient metric from the ordinary orbit path metric, which is harmonic only under a stated constant-stabilizer condition.

Sources

• Local UnsolvedMath registry, permanent ID $20001005$, source code catalogue entry; aim-combinatorics-notes.json, notes[129], tag problem.
• 2013 AIM workshop "Generalizations of chip-firing and the critical group," Problem 40, Jordan Ellenberg, with Matt Baker's metric-graph comment: https://aimath.org/pastworkshops/chipfiringproblems.pdf. Statement status exact. No PDF was accessed for this draft.
• Baker's paper, tropical cone-stack construction, and harmonic-quotient results have no full bibliographic citations in the supplied record. No original remarks or literature field are supplied.
• Research attempt 1: partial_result, status partially_solved, classification PARTIAL-PROGRESS, rigor label proved, novelty confidence low. Created and updated 14 August 2026. This machine-generated aid is not peer reviewed; independently verify before citation.
• Ulam AI / UnsolvedMath Contributors; CC BY 4.0.