Pointed Gromov–Hausdorff Limit Graphs

• Pointed Gromov–Hausdorff limit graphs are infinite rooted graphs obtained by recursive approximation where every fixed-radius ball eventually stabilizes.
• They enable isomorphism classification via dihedral symmetry on vertex addresses, linking local combinatorial structure with metric convergence without global distortion control.
• The framework underpins a detailed horofunction compactification, distinguishing Busemann from non-Busemann boundary points and paralleling features in Sierpiński carpet models.

Pointed Gromov–Hausdorff limit graphs are rooted infinite graphs obtained as limits of sequences of pointed finite graphs in such a way that every fixed-radius ball around the distinguished vertex eventually becomes stable. In the graph setting, this viewpoint turns local combinatorial stabilization into a metric notion of convergence, and it supports finer asymptotic constructions such as isomorphism classification, metric compactification, and horofunction boundaries. A central recent instance is the family of Sierpinski polygon limit graphs determined by infinite words over $r$ letters, with $r\ge 1$ not divisible by $4$, where the finite approximants converge strongly in the pointed Gromov–Hausdorff sense to an infinite rooted graph $\Gamma_\xi^{(r)}$ (D'Angeli et al., 31 Jul 2025).

1. Pointed Gromov–Hausdorff convergence in rooted graphs

For boundedly-compact pointed metric spaces $(X,o)$, the pointed Gromov–Hausdorff metric compares large balls around the distinguished points and lets the radius tend to infinity. In the specialization to locally finite rooted graphs, the vertex set is viewed as a discrete metric space with the graph-distance

$d_G(u,v)=\text{length of shortest path from %%%%5%%%% to %%%%6%%%%}.$

Because balls of finite radius contain only finitely many vertices, each rooted graph $(G,o)$ is a boundedly-compact pointed metric space. In this setting, pointed Gromov–Hausdorff convergence is equivalent to rooted-ball stabilization: for every fixed radius $r$, the balls $B_{G_n}(o_n,r)$ are eventually isomorphic to $B_G(o,r)$ (Khezeli, 2018).

The Sierpinski polygon construction uses a stronger formulation. A sequence of pointed graphs $r\ge 1$0 strongly converges to $r\ge 1$1 if for every integer $r\ge 1$2 there is $r\ge 1$3 such that whenever $r\ge 1$4 there is a pointed graph-isometry

$r\ge 1$5

sending $r\ge 1$6. In that framework, strong convergence immediately implies the usual pointed Gromov–Hausdorff convergence of the underlying metric spaces, so no explicit computation of a Gromov–Hausdorff distance is required (D'Angeli et al., 31 Jul 2025).

A recurrent misconception is that one must control a global metric distortion directly. In these graph families, the decisive input is instead the eventual isometry type of finite balls around the root. This is precisely the mechanism that allows recursive graph constructions to pass to infinite pointed limits.

2. Recursive construction of Sierpinski polygon approximants

The Sierpinski polygon graphs are defined from the cyclic graph $r\ge 1$7 on

$r\ge 1$8

equipped with its usual graph-metric $r\ge 1$9. The parameters

$4$0

govern the gluing rule. The base graph is $4$1. Inductively, assuming $4$2 has been constructed, one forms $4$3 disjoint labeled copies $4$4, $4$5, writes vertices in the $4$6-th copy as words of length $4$7 ending in $4$8, and then performs a prescribed pairwise identification between neighboring copies (D'Angeli et al., 31 Jul 2025).

More concretely, for each $4$9, the vertex

$\Gamma_\xi^{(r)}$0

is identified with

$\Gamma_\xi^{(r)}$1

and an analogous “opposite” vertex of $\Gamma_\xi^{(r)}$2 is glued to $\Gamma_\xi^{(r)}$3. The resulting connected graph is $\Gamma_\xi^{(r)}$4. An infinite sequence

$\Gamma_\xi^{(r)}$5

determines a distinguished vertex $\Gamma_\xi^{(r)}$6 in $\Gamma_\xi^{(r)}$7, so each finite approximant is regarded as a pointed graph $\Gamma_\xi^{(r)}$8 (D'Angeli et al., 31 Jul 2025).

This recursive definition supplies a canonical address system for vertices and an equally canonical choice of roots. A plausible implication is that the tail of the address $\Gamma_\xi^{(r)}$9 should encode the asymptotic position of the root inside the self-similar graph, and that implication is borne out by the isomorphism classification.

3. Limit graphs and classification by dihedral symmetry

For every infinite word $(X,o)$0, the recursive construction yields a strong pointed Gromov–Hausdorff limit

$(X,o)$1

where $(X,o)$2 is the infinite Sierpinski polygon limit graph determined by $(X,o)$3 (D'Angeli et al., 31 Jul 2025). The limit graph is always pointed at the infinite word $(X,o)$4, and its finite-radius neighborhoods are inherited from sufficiently deep finite approximants.

The classification of these limit graphs is governed by the symmetries of the base cycle. The full isometry group of $(X,o)$5 is the dihedral group $(X,o)$6 of order $(X,o)$7, acting on $(X,o)$8 by permutations that preserve cyclic adjacency. This action extends coordinate-wise to infinite sequences in $(X,o)$9. For every $d_G(u,v)=\text{length of shortest path from %%%%5%%%% to %%%%6%%%%}.$0 and every $d_G(u,v)=\text{length of shortest path from %%%%5%%%% to %%%%6%%%%}.$1, relabeling the $d_G(u,v)=\text{length of shortest path from %%%%5%%%% to %%%%6%%%%}.$2 copies induces a pointed-graph isomorphism

$d_G(u,v)=\text{length of shortest path from %%%%5%%%% to %%%%6%%%%}.$3

and hence a pointed isomorphism of the limits

$d_G(u,v)=\text{length of shortest path from %%%%5%%%% to %%%%6%%%%}.$4

Two sequences $d_G(u,v)=\text{length of shortest path from %%%%5%%%% to %%%%6%%%%}.$5 are called cofinal, written $d_G(u,v)=\text{length of shortest path from %%%%5%%%% to %%%%6%%%%}.$6, if they agree from some index onward. If $d_G(u,v)=\text{length of shortest path from %%%%5%%%% to %%%%6%%%%}.$7, then $d_G(u,v)=\text{length of shortest path from %%%%5%%%% to %%%%6%%%%}.$8 and $d_G(u,v)=\text{length of shortest path from %%%%5%%%% to %%%%6%%%%}.$9 represent the same vertex of the limit graph, so $(G,o)$0. Conversely, any pointed isomorphism $(G,o)$1 comes from some $(G,o)$2 (D'Angeli et al., 31 Jul 2025).

The resulting theorem is exact: $(G,o)$3 as un-pointed graphs if and only if there exists $(G,o)$4 such that

$(G,o)$5

Thus the unpointed isomorphism class is determined by the tail of the address modulo dihedral symmetry. This reduces a graph-theoretic classification problem to tail-equivalence in a symbolic space together with a finite group action.

4. Horofunction boundaries, Busemann points, and non-Busemann points

The asymptotic geometry of a pointed limit graph can be refined through the horofunction compactification. For any locally finite connected graph $(G,o)$6 with graph-metric $(G,o)$7, after fixing a base vertex $(G,o)$8, one defines for each $(G,o)$9 the distance-normalized function

$r$0

The closure of $r$1 in $r$2 is the horofunction compactification, and the horofunction boundary is

$r$3

A horofunction is a Busemann point if it arises as the pointwise limit of an almost-geodesic ray; otherwise it is non-Busemann. In the specialized form used here, every element of $r$4 is the pointwise limit of some weakly-geodesic ray, and every weakly-geodesic ray defines a horofunction (D'Angeli et al., 31 Jul 2025).

In the Sierpinski polygon limit graphs, the relevant large-scale obstructions are the “holes” inherited from finite levels. Roughly speaking, each $r$5 has one large central hole whose boundary is an embedded generalized Koch-curve. Gluing copies around that hole produces two canonical sequences of cut-vertices $r$6 and $r$7 receding to infinity along two distinct geodesic rays. When the defining sequence is eventually constant, say $r$8, the boundary admits an explicit classification: exactly two Busemann points occur, namely those determined by the two gluing geodesic rays through $r$9 and $B_{G_n}(o_n,r)$0, and there are countably many non-Busemann points arising, for example, from antipodal sequences of vertices on opposite sides of each hole and from their integer-shifts along the hole’s perimeter (D'Angeli et al., 31 Jul 2025).

The distinction is structural rather than terminological. In these graphs, not every point at infinity is seen by an almost-geodesic ray. The paper constructs, for each $B_{G_n}(o_n,r)$1, a weakly-geodesic ray $B_{G_n}(o_n,r)$2 visiting the midpoint of the $B_{G_n}(o_n,r)$3-th hole at offset $B_{G_n}(o_n,r)$4, proves convergence to a horofunction, and shows that only two distinct accumulation points have the almost-geodesic property. This provides a concrete instance in which Busemann and non-Busemann boundary points coexist in a single pointed Gromov–Hausdorff limit graph.

5. Comparison with infinite Sierpiński carpet graphs

A closely related model is the infinite Sierpiński carpet graph $B_{G_n}(o_n,r)$5 associated with an infinite word

$B_{G_n}(o_n,r)$6

where $B_{G_n}(o_n,r)$7 and $B_{G_n}(o_n,r)$8. The finite rooted approximants $B_{G_n}(o_n,r)$9 are built by starting from the $B_G(o,r)$0-cycle $B_G(o,r)$1, then at each step taking eight disjoint copies of $B_G(o,r)$2, placing them in the $B_G(o,r)$3 Sierpiński carpet pattern with the central piece removed, and choosing the new root in the copy labeled $B_G(o,r)$4. The infinite rooted graph $B_G(o,r)$5, denoted $B_G(o,r)$6 in the paper, is the union of these approximants glued compatibly along roots (D'Angeli et al., 2015).

The pointed Gromov–Hausdorff convergence mechanism is the same as in the polygon case: for every radius $B_G(o,r)$7, once $B_G(o,r)$8 is large enough,

$B_G(o,r)$9

so the limit follows by matching finite balls exactly. The resulting infinite graph is locally finite and connected, and it admits an explicit horofunction compactification. Its boundary is described in terms of geodesic, almost-geodesic, and weakly-geodesic rays, with the classification depending on which symbols $r\ge 1$00 occur infinitely often in the address. Under the product Bernoulli measure giving each letter equal probability $r\ge 1$01 or $r\ge 1$02, the boundary consists with probability $r\ge 1$03 of exactly four Busemann points and countably many non-Busemann points (D'Angeli et al., 2015).

The comparison is instructive. Both the polygon and carpet families produce pointed Gromov–Hausdorff limit graphs from recursive self-similar approximants; both admit a full horofunction analysis; and both exhibit non-Busemann points generated by large detours around recursively created holes. The precise cardinalities differ, but the qualitative mechanism is parallel.

6. Broader GH- and GHP-limit context

Pointed Gromov–Hausdorff limit graphs sit within a broader theory of metric limits with or without additional structure. A unified framework for Gromov–Hausdorff-type metrics treats compact or boundedly-compact metric spaces equipped with a measure, a point, a closed subset, a curve, or tuples of such data, and proves completeness and separability under suitable conditions. In the graph specialization, rooted local weak convergence agrees with pointed Gromov–Hausdorff convergence once the graph is viewed through its graph-distance and, when necessary, its edge relation as extra structure (Khezeli, 2018).

Related work on degenerating flat Riemannian surfaces shows that pointed Gromov–Hausdorff limits need not arise only from discrete graph approximants. For a one-parameter family of flat genus-$r\ge 1$04 curves in the maximally collapsed case, after rescaling the flat metric on each fiber to unit diameter, the pointed Gromov–Hausdorff limit is a metric graph $r\ge 1$05 obtained from the dual intersection complex by collapsing away the complement of the minimality locus of the Kontsevich–Soibelman weight function. In that setting, every positive genus may occur as a collapsed limit (Sustretov, 2018).

A different regime appears in random unlabelled $r\ge 1$06-trees. There the global limit is not a locally finite infinite graph but the Brownian Continuum Random Tree: the random unlabelled $r\ge 1$07-tree $r\ge 1$08, equipped with graph-distance and the uniform vertex-measure, converges after rescaling by $r\ge 1$09 in the Gromov–Hausdorff–Prokhorov topology to the CRT. At the same time, the neighborhood of a uniform random vertex out to radius $r\ge 1$10 converges in total variation to the corresponding neighborhood in an infinite random $r\ge 1$11-tree $r\ge 1$12. This separates the local rooted limit from the global macroscopic scaling limit (Jin et al., 2018).

Taken together, these developments show that pointed Gromov–Hausdorff limit graphs occupy several distinct roles. In deterministic recursive graph families such as Sierpinski polygons and carpets, they encode exact local stabilization and support explicit boundary theory. In geometric degeneration, they arise as collapsed metric graphs with lengths determined by algebro-geometric data. In random combinatorics, they interface with Gromov–Hausdorff–Prokhorov scaling limits and local weak limits, clarifying the difference between local graph structure and global metric shape.