On Geometric Models of String Algebras: Uniqueness of Surfaces and Existence of Red Punctures

Abstract: A geometric model for string algebras was recently established in \cite{BC24}. Building upon this framework, we characterize the class of string algebras whose geometric models are unique up to equivalence of labelled tiled surfaces. Moreover, we provide a necessary and sufficient condition for all geometric models of a string algebra to be entirely free of red punctures, and further give a combinatorial description of string algebras with a common red puncture across all geometric models. In addition, we derive a sufficient condition for a string algebra guaranteeing the presence of red punctures in all geometric models.

• The paper characterizes model uniqueness through three local conditions on Type (I) and Type (III) non-gentle vertices, reducing a global surface problem to finite quiver checks.
• The paper proves that all models are puncture-free exactly when every primitive oriented cycle is gentle-bounded, while essential cycles precisely characterize red punctures shared by every model.
• The paper shows that essential, shuttle, or multi-coupled cycles guarantee red punctures in every saturated model, but leaves open whether these structures are also necessary.

Background and setting

The geometric model of Baur and Coelho Simões realizes a string algebra $A = \mathbf{k}Q/I$ as a labelled tiled surface: a marked surface equipped with a dissection into polygons by red-arcs, together with labels encoding the relations of length at least three that distinguish string algebras from locally gentle algebras (Baur et al., 2024). The construction proceeds through a locally gentle cover $\mathbf{k}Q/J$ of $A$, where $J \subseteq I$ is a quadratic ideal for which the quotient is locally gentle; the surface is built from dual cellular dissections in the sense of Opper–Plamondon–Schroll (Opper et al., 2018) and Palu–Pilaud–Plamondon. Because a string algebra may admit several distinct maximal locally gentle covers, its labelled tiled surface is not unique in general, and different models of the same algebra may differ in whether they carry red punctures.

This paper restricts attention to saturated labelled tiled surfaces, those arising from saturated locally gentle covers defined via maximal quadratic ideals, and addresses two questions: when is the geometric model of a string algebra unique up to equivalence of labelled tiled surfaces, and when do all geometric models share (or avoid) red punctures.

Non-gentle vertices and admissible deletion data

The source of non-uniqueness is localized at non-gentle vertices, which the paper classifies into three types:

• Type (I): $v$ has two incoming arrows $a,b$ and two outgoing arrows $c,d$ with all four compositions $ac,ad,bc,bd$ lying in $I$.
• Type (II): exactly three of these four compositions lie in $I$.
• Type (III): $\mathbf{k}Q/J$0 has degree three, with either two incoming arrows $\mathbf{k}Q/J$1, one outgoing arrow $\mathbf{k}Q/J$2, and $\mathbf{k}Q/J$3 (Type III.a), or the symmetric configuration (Type III.b).

Transforming a non-gentle vertex into a gentle vertex requires deleting relations: Type (II) admits a unique deletion choice, while Types (I) and (III) each admit exactly two. Consequently, if $\mathbf{k}Q/J$4 count the non-gentle vertices of Types (I) and (III), there are up to $\mathbf{k}Q/J$5 distinct locally gentle covers, and hence up to $\mathbf{k}Q/J$6 geometric models. The paper also records a structural fact about loops: a vertex of a string algebra carries at most two loops, and if it carries two, no other arrows are incident to it.

The central combinatorial device is the admissible deletion datum $\mathbf{k}Q/J$7: a choice of 2-relations to delete at every non-gentle vertex, together with deletion of all generating relations of length at least three, yielding a saturated locally gentle cover $\mathbf{k}Q/J$8. The deleted relations correspond to labels $\mathbf{k}Q/J$9 on the surface. Two deletion data are equivalent if a bound-quiver isomorphism preserves the label data in a local-symmetry-compatible sense. A key lemma establishes a bijection between equivalence classes of geometric models of $A$0 and equivalence classes $A$1 of admissible deletion data, so uniqueness of the model reduces to transitivity of the label-preserving automorphism group on $A$2.

Uniqueness of the geometric model

The main theorem characterizes uniqueness as follows: the geometric model of $A$3 is unique if and only if every non-gentle vertex of Type (I) or Type (III) satisfies three conditions:

1. (U1) For a Type (I) vertex $A$4: (a1) there exists a label-preserving bound-quiver isomorphism between one of the symmetric pairs of branches at $A$5; (a2) the paired branches lie in distinct connected components after removing $A$6.
2. (U2) For a Type (III) vertex: the two relevant branches admit a label-preserving bound-quiver isomorphism.
3. (U3) No pair of Type (III) non-gentle vertices occurs in one of four forbidden interacting configurations.

The sufficiency proof introduces a directed dependency graph $A$7 on the Type (I)/(III) vertices, with an edge $A$8 when changing the deletion choice at $A$9 can destroy the global extendability of the local symmetry at $J \subseteq I$0. An analysis of local connection patterns shows every dependency edge is single-branch (giving strict inclusion $J \subseteq I$1) or double-branch, with genuine double-branch edges only from Type (III) to Type (I). Any directed cycle would force either a closed chain of strict inclusions, a violation of condition (a2), or one of the configurations excluded by (U3); hence $J \subseteq I$2 is acyclic. Induction on the number of such vertices, peeling off sinks, then glues local symmetries to global label-preserving isomorphisms, showing all deletion data are equivalent.

The converse is proved by contraposition: failure of any condition produces two inequivalent deletion data, distinguished by an invariant called the permitted-connection graph of the distinguished branches, whose connected-component partition differs under the two local deletion choices. Notably, when all non-gentle vertices are of Type (II), the model is absolutely unique — not merely up to equivalence — since $J \subseteq I$3 is a singleton.

The practical consequence is checkable: uniqueness is decidable from finitely many local quiver configurations around non-gentle vertices, without constructing any surface.

Geometric models without red punctures

Red punctures correspond bijectively to cyclic equivalence classes of primitive permitted cycles of the underlying locally gentle cover. The paper defines a primitive oriented cycle $J \subseteq I$4 to be gentle-bounded if some consecutive subpath $J \subseteq I$5 lies in $J \subseteq I$6 and the middle vertex $J \subseteq I$7 is either gentle or a Type (II) non-gentle vertex whose deleted relation is not $J \subseteq I$8. The characterization is clean:

All geometric models of $J \subseteq I$9 contain no red puncture if and only if every primitive oriented cycle in $v$0 is gentle-bounded.

Sufficiency follows because each cycle then contains a 2-subpath surviving in every $v$1, so no cover admits a permitted cycle; since $v$2 is finite, every $v$3 is finite-dimensional, and finite-dimensional locally gentle algebras have puncture-free models. Necessity is subtler: given a non-gentle-bounded cycle of minimal length, the paper constructs a deletion datum making it a permitted cycle. The construction requires showing the cycle has no repeated arrow (via a minimal-subwalk argument), and that local deletion requirements at the non-gentle vertices traversed by the cycle are mutually compatible — which holds because degree bounds force each vertex to be traversed at most twice, and the two passages through a Type (I) vertex determine exactly one valid local choice. Acyclic quivers trivially satisfy the condition.

An important caveat stated explicitly: a cycle failing to be gentle-bounded only guarantees a red puncture in at least one model, not in all; conversely, a gentle-bounded cycle cannot form a complete fan of a puncture but may still participate in one whose complete fan comes from another cycle.

Geometric models with red punctures

For the opposite direction, the paper introduces three subquiver structures guaranteeing punctures in every model.

Essential cycles. A vertex on a cycle is stable if it is gentle with $v$4, or a Type (II) non-gentle vertex in one of two specific local configurations (one where $v$5 and is precisely the uniquely deleted relation, one where $v$6). A cycle all of whose vertices are stable is an essential cycle. Such a cycle is $v$7-permitted for every deletion datum, so every model contains a red puncture. Moreover, this condition is necessary and sufficient for a common red puncture: all geometric models share a common red puncture exactly when $v$8 contains an essential cycle. The necessity argument rules out gentle vertices carrying relations on the cycle and Type (I)/(III) vertices, since at the latter one can always choose a local deletion killing the cycle's permissibility.

Shuttle cycles. A shuttle cycle is a chain of antiparallel arrow pairs between distinct vertices $v$9 ($a,b$0), with stable endpoints and intermediate vertices gentle or of Types (I)/(II), at least one being non-gentle. Regardless of the deletion choices, every arrow in the chain has a unique permitted successor within the subquiver, producing an infinite non-zero path and hence a puncture in every model. The degenerate cases $a,b$1 (two loops at one vertex) and $a,b$2 (an essential cycle) are excluded deliberately.

Multi-coupled cycles. These generalize coupled cycles — pairs of primitive oriented cycles sharing a path of vertices, traversed in the same or opposite directions, with at least one shared non-gentle vertex of Type (I) or (II). Multiple coupled skeletons with pairwise disjoint common regions are formally contracted to marked vertices, and the assembly is a multi-coupled cycle when all induced cycles are relatively essential and the skeletons are chained by shared induced cycles. A technical lemma shows that within a coupled skeleton, every incoming arrow extends either out of the region or to a primitive permitted cycle inside it; iterating this over the finite arrow set yields a primitive permitted cycle for every saturated cover, hence a puncture in every model.

Combining these: if $a,b$3 contains an essential cycle, a shuttle cycle, or a multi-coupled cycle, then every geometric model contains a red puncture. The paper does not claim necessity of this disjunction and poses it as an open question. Partial evidence is provided by a lemma showing that if every cover contains some primitive permitted cycle, then one can always be chosen avoiding all Type (III) non-gentle vertices; the proof rests on a local observation that two covers differing only at a Type (III) vertex cannot both have permitted cycles through that vertex, followed by induction over such vertices.

Examples

Seven worked examples illustrate the theory. They include: an acyclic quiver with four pairwise-isomorphic covers whose models nonetheless split into two inequivalence classes due to differing label positions; a Type (I) vertex violating (U1)(a2), giving two non-isomorphic covers; two Type (III) vertices violating (U3); a two-vertex quiver where some covers are infinite-dimensional (yielding punctured models) and others finite-dimensional (puncture-free models); a genus-one example where equivalent covers produce inequivalent surfaces; and examples with multiple red punctures whose counts differ across models. These confirm both directions of the uniqueness criterion and demonstrate that the number of red punctures is itself model-dependent.

Limitations and open questions

Several boundaries of the results deserve emphasis. First, everything is formulated for saturated labelled tiled surfaces; behavior of non-saturated models is outside scope. Second, the sufficient condition of Theorem thm: all-red puncture (presence of an essential, shuttle, or multi-coupled cycle) is not known to be necessary; the paper leaves open whether every string algebra whose models all carry red punctures must contain one of these structures, offering only the Type-(III)-elimination lemma as supporting evidence. Third, the classification counts models up to $a,b$4 but does not give a closed formula for the number of equivalence classes of geometric models in the non-unique case. Finally, the correspondence between red punctures and permitted cycles relies on the surface framework of Palu–Pilaud–Plamondon, so the results inherit whatever hypotheses that framework imposes on the marked surfaces involved.

Conclusion

The paper reduces two global questions about labelled tiled surfaces of string algebras to purely local, checkable conditions on the bound quiver. Uniqueness of the geometric model is characterized by symmetry and connectivity conditions at Type (I) and (III) non-gentle vertices, proved via an acyclicity argument on a dependency graph. Absence of red punctures in all models is equivalent to all primitive oriented cycles being gentle-bounded, while common red punctures are exactly detected by essential cycles, and their presence in all models is ensured by essential, shuttle, or multi-coupled cycles. The remaining gap — necessity of the latter sufficient condition — is identified precisely and left as an explicit question.