Coned-off Cayley Graphs in Small Cancellation

• Coned-off Cayley graphs are defined by enlarging the generating set so that entire relators become cliques, producing a hyperbolic model.
• They encode relator traversal by decomposing geodesics into segments within relators and isolated edges, revealing the underlying combinatorics.
• The construction enables a coding of the hyperbolic boundary that leads to hyperfinite boundary actions under graphical small cancellation conditions.

Searching arXiv for the cited paper and closely related work on coned-off Cayley graphs in graphical small cancellation. First, I’ll look up the main paper by arXiv id (Karpinski et al., 6 Sep 2025). Coned-off Cayley graphs are hyperbolic models attached to groups whose ordinary Cayley graphs retain large non-hyperbolic substructures. In the setting of graphical small cancellation, the coned-off Cayley graph is not an auxiliary construction but the central geometric object: starting from a graphical presentation $G(\Gamma)$, one enlarges the generating set so that entire embedded relators become uniformly bounded in diameter, producing a graph $Y$ whose Gromov boundary $\partial Y$ supports the boundary action studied in "Graphical small cancellation and hyperfiniteness of boundary actions" (Karpinski et al., 6 Sep 2025). In that framework, the coned-off graph packages the geometry of relator traversal, provides the hyperbolic boundary, and supplies the combinatorial structure used to prove that the induced orbit equivalence relation on $\partial Y$ is hyperfinite for a class of graphical $C'(1/10)$ groups including infinitely presented classical small cancellation groups (Karpinski et al., 6 Sep 2025).

1. Definition from graphical small cancellation presentations

The construction begins with a set $S$ and an $S$-labeled graph $\Gamma$, where labels lie in $S \coprod S^{-1}$, compatibly with edge orientation. The associated graphical presentation is

$G(\Gamma)=\langle S \mid \ell(\gamma) : \text{%%%%9%%%% is a simple closed path in %%%%10%%%%}\rangle.$

A path $Y$0 is a piece if there is another path $Y$1 with $Y$2 but no label-preserving graph automorphism of $Y$3 sends $Y$4 to $Y$5. The graph satisfies the graphical $Y$6 condition if no two edges with the same initial vertex have the same label and every piece $Y$7 in a component $Y$8 satisfies

$Y$9

These notions are the small-cancellation input from which the coned-off geometry is defined (Karpinski et al., 6 Sep 2025).

For $\partial Y$0 graphical small cancellation groups, each connected component $\partial Y$1 embeds isometrically and convexly into the ordinary Cayley graph

$\partial Y$2

and these embedded copies are called relators in $\partial Y$3. A simple closed path in $\partial Y$4 contained in a relator is called a contour. The coned-off construction is built from this embedded-relator picture rather than from peripheral cosets or additional apex vertices (Karpinski et al., 6 Sep 2025).

The paper defines

$\partial Y$5

and then sets

$\partial Y$6

This is called the coned-off graph of $\partial Y$7 (Karpinski et al., 6 Sep 2025). The additional generators are therefore exactly the group elements represented by labels of arbitrary paths in the defining graph $\partial Y$8. Geometrically, whenever two vertices of $\partial Y$9 lie in a common embedded relator, the path between them inside that relator determines an element of $\partial Y$0, so they become adjacent in $\partial Y$1. The paper states that one may think of $\partial Y$2 as obtained from $\partial Y$3 by replacing every relator $\partial Y$4 in $\partial Y$5 by the complete subgraph on its vertices; this yields the same metric space as $\partial Y$6 (Karpinski et al., 6 Sep 2025).

This model differs from the more standard relative-hyperbolic coned-off graph. In the latter, one typically adds a cone-vertex for each left coset of a peripheral subgroup. Here, by contrast, one “cones off” each embedded relator by turning its vertex set into a clique. The philosophy is analogous—collapse large non-hyperbolic pieces to expose a hyperbolic large-scale structure—but the mechanism is specific to graphical small cancellation (Karpinski et al., 6 Sep 2025).

2. Hyperbolicity and the relator-crossing metric

A central fact is that the coned-off Cayley graph $\partial Y$7 is hyperbolic. The paper states that, by a theorem of Gruber–Sisto, for $\partial Y$8 graphical small cancellation groups, the coned-off Cayley graph $\partial Y$9 is always hyperbolic; the main theorem assumes the stronger hypothesis $C'(1/10)$0, but hyperbolicity already follows at the $C'(1/10)$1 level (Karpinski et al., 6 Sep 2025). Thus the coned-off graph provides the hyperbolic space whose boundary is later studied.

The geometry of $C'(1/10)$2 is encoded through the way geodesics in $C'(1/10)$3 pass through relators. The paper summarizes this by saying that the metric $C'(1/10)$4 “counts how many relators any geodesic in $C'(1/10)$5 between two given points passes through” (Karpinski et al., 6 Sep 2025). More precisely, if $C'(1/10)$6 are vertices in $C'(1/10)$7, $C'(1/10)$8 is a geodesic in $C'(1/10)$9 from $S$0 to $S$1, and $S$2, then $S$3 is the minimal number such that

$S$4

where each $S$5 is either a path in some relator in $S$6 or an edge in $S$7 not occurring on any relator (Karpinski et al., 6 Sep 2025).

This decomposition gives the correct combinatorial model for $S$8-geodesics. A geodesic in the coned-off metric is not primarily measuring ordinary word length in $S$9; instead, it records passage through a sequence of relators or isolated non-relator edges. A plausible implication is that $S$0 suppresses internal geometry within relators while preserving the large-scale combinatorics of how relators intersect and how paths move from one relator to another. That is precisely the feature needed to recover a hyperbolic “skeleton” from a non-hyperbolic ambient Cayley graph.

The paper emphasizes that $S$1 is connected and geodesic because it is a Cayley graph, and that it need not be locally finite, since adding all of $S$2 can create infinitely many edges at a vertex (Karpinski et al., 6 Sep 2025). Accordingly, the analysis does not rely on local finiteness of $S$3. Instead, it uses geodesic rays in the locally finite graph $S$4 while measuring escape to infinity using the coned-off metric $S$5. That methodological shift is one of the paper’s defining features (Karpinski et al., 6 Sep 2025).

3. Boundary theory for the coned-off graph

The boundary under study is the Gromov boundary of $S$6. For a hyperbolic metric space $S$7, the Gromov product is

$S$8

A sequence $S$9 converges to infinity if $\Gamma$0, and $\Gamma$1 is the space of such sequences modulo the equivalence relation $\Gamma$2 defined by $\Gamma$3. The paper applies this sequential definition to $\Gamma$4 (Karpinski et al., 6 Sep 2025).

Because $\Gamma$5 is hyperbolic and has countable vertex set, its boundary $\Gamma$6 is Polish; the paper attributes this to Oyakawa’s lemma on countable hyperbolic graphs (Karpinski et al., 6 Sep 2025). The group acts on $\Gamma$7 by left multiplication, since $\Gamma$8 is a Cayley graph of $\Gamma$9 with generating set $S \coprod S^{-1}$0. This action preserves graph distance and therefore induces an isometric action on $S \coprod S^{-1}$1 (Karpinski et al., 6 Sep 2025). The dynamical object of interest is exactly this boundary action

$S \coprod S^{-1}$2

A key issue is that a geodesic ray in the ordinary Cayley graph $S \coprod S^{-1}$3 need not determine a point of $S \coprod S^{-1}$4. The paper introduces $S \coprod S^{-1}$5, the set of geodesic rays in $S \coprod S^{-1}$6 based at $S \coprod S^{-1}$7 that represent $S \coprod S^{-1}$8, and proves that $S \coprod S^{-1}$9 for every $G(\Gamma)=\langle S \mid \ell(\gamma) : \text{%%%%9%%%% is a simple closed path in %%%%10%%%%}\rangle.$0 (Karpinski et al., 6 Sep 2025). More precisely, for a geodesic ray $G(\Gamma)=\langle S \mid \ell(\gamma) : \text{%%%%9%%%% is a simple closed path in %%%%10%%%%}\rangle.$1 in $G(\Gamma)=\langle S \mid \ell(\gamma) : \text{%%%%9%%%% is a simple closed path in %%%%10%%%%}\rangle.$2, the following are equivalent:

1. there exists $G(\Gamma)=\langle S \mid \ell(\gamma) : \text{%%%%9%%%% is a simple closed path in %%%%10%%%%}\rangle.$3 such that $G(\Gamma)=\langle S \mid \ell(\gamma) : \text{%%%%9%%%% is a simple closed path in %%%%10%%%%}\rangle.$4;
2. $G(\Gamma)=\langle S \mid \ell(\gamma) : \text{%%%%9%%%% is a simple closed path in %%%%10%%%%}\rangle.$5;
3. there are vertices $G(\Gamma)=\langle S \mid \ell(\gamma) : \text{%%%%9%%%% is a simple closed path in %%%%10%%%%}\rangle.$6 on $G(\Gamma)=\langle S \mid \ell(\gamma) : \text{%%%%9%%%% is a simple closed path in %%%%10%%%%}\rangle.$7 such that each segment $G(\Gamma)=\langle S \mid \ell(\gamma) : \text{%%%%9%%%% is a simple closed path in %%%%10%%%%}\rangle.$8 either lies in a relator or is a non-relator edge, and $G(\Gamma)=\langle S \mid \ell(\gamma) : \text{%%%%9%%%% is a simple closed path in %%%%10%%%%}\rangle.$9 is a geodesic ray in $Y$00 (Karpinski et al., 6 Sep 2025).

This equivalence identifies the coned-off metric as the correct notion of escape to infinity. A ray in $Y$01 defines a boundary point of $Y$02 precisely when it makes unbounded progress after relators have been collapsed. The paper further proves that the geodesic and sequential boundaries coincide,

$Y$03

which permits boundary points to be coded by geodesic rays (Karpinski et al., 6 Sep 2025).

4. Geodesic bundles and the internal structure of $Y$04

The proof theory surrounding the coned-off graph depends on a detailed analysis of geodesic bundles $Y$05. The existence of $Y$06-geodesic representatives for boundary points is obtained by combining hyperbolicity of $Y$07 with Strebel’s classification of geodesic bigons and triangles in small cancellation Cayley graphs (Karpinski et al., 6 Sep 2025). Starting from a sequence $Y$08 in $Y$09, one considers $Y$10-geodesics from $Y$11 to $Y$12 and proves that, for each coned-off distance level $Y$13, the set of vertices on those geodesics with $Y$14 is finite. The finiteness arises from the fact that hyperbolicity in $Y$15 constrains those vertices to lie either on a fixed reference geodesic or on contours common to two geodesics, while small cancellation bounds where such contours can occur (Karpinski et al., 6 Sep 2025).

Once geodesic representatives are available, the paper derives a strong fellow-travel property. If $Y$16, then they can be decomposed into alternating common subsegments and contour-bounded bigons (Karpinski et al., 6 Sep 2025). More precisely, there are vertex sequences $Y$17 on $Y$18 and $Y$19 on $Y$20 such that each corresponding pair of segments either coincides or, together with two short connecting paths, bounds a contour. Moreover, once a decomposition of $Y$21 into $Y$22-edges via relators $Y$23 is fixed, every other $Y$24 lies in $Y$25 and intersects every $Y$26 (Karpinski et al., 6 Sep 2025).

This is a rigid visibility statement for geodesic bundles in the coned-off geometry. A plausible implication is that the relator-chain associated to one representative ray largely determines the geometry of all representatives of the same boundary point. Such rigidity substitutes for more classical bounded-ray-bundle properties familiar from hyperbolic groups and buildings, but the substitute is formulated entirely in terms of relators and contours rather than ambient local finiteness (Karpinski et al., 6 Sep 2025).

The paper then encodes each ray by its label sequence

$Y$27

Since $Y$28 is finite and discrete, $Y$29 is compact metrizable. The bundle $Y$30 is compact because, for each coned-off distance level $Y$31, all vertices on all rays in $Y$32 at level $Y$33 lie in the finite union $Y$34 (Karpinski et al., 6 Sep 2025). This compactness implies that $Y$35 has a lexicographically least element after fixing an order on $Y$36.

5. Coding boundary points and hyperfinite boundary actions

The lexicographically least representative yields a coding map

$Y$37

where $Y$38 is the label of the lexicographically least geodesic ray in $Y$39 (Karpinski et al., 6 Sep 2025). The set

$Y$40

is shown to be closed, and from this the map $Y$41 is proved Borel and injective (Karpinski et al., 6 Sep 2025). The boundary is therefore embedded into the shift space $Y$42 through a coding that is built directly from coned-off geodesic geometry.

The dynamical argument then compares the orbit equivalence relation $Y$43 on $Y$44 with tail equivalence on $Y$45. Let $Y$46 denote tail equivalence on $Y$47, defined by

$Y$48

The paper pulls this back along $Y$49 to obtain

$Y$50

and then defines

$Y$51

Since $Y$52 is hyperfinite, $Y$53 is hyperfinite as well (Karpinski et al., 6 Sep 2025).

The decisive remaining step is a finite-index statement. Under the hypothesis that $Y$54 is extremely fine—meaning that there exists $Y$55 such that every edge of $Y$56 lies in at most $Y$57 simple closed paths—the paper proves that there exists $Y$58 such that each $Y$59-class in $Y$60 contains at most $Y$61 $Y$62-classes (Karpinski et al., 6 Sep 2025). The proof is explicitly relator-by-relator. One fixes a geodesic ray $Y$63 and its associated sequence of relators $Y$64; any translated lexicographically least representative of an orbit-equivalent boundary point eventually lives in the same chain of relators. Extreme fineness bounds the number of possible entry points into each $Y$65, because a ray can enter either along $Y$66 or through the last contour in $Y$67 that it traverses, and extreme fineness bounds the number of such contours through a chosen edge (Karpinski et al., 6 Sep 2025). A pigeonhole argument at two far-apart relators then forces two rays to share the same entry points, and because both are lexicographically least, the intervening segment must coincide; their labels are therefore tail equivalent (Karpinski et al., 6 Sep 2025).

From the finite bound on the number of tail classes per orbit, the paper concludes hyperfiniteness of the boundary action using a proposition of Jackson–Kechris–Louveau (Karpinski et al., 6 Sep 2025). The main theorem states:

$Y$68

with $Y$69 finite, $Y$70 having countably many finite connected components, and $Y$71 extremely fine. Let $Y$72 denote the associated coned-off Cayley graph. Then

$Y$73

is hyperfinite (Karpinski et al., 6 Sep 2025).

6. Classical small cancellation as a special case

The paper emphasizes that classical small cancellation groups are exactly the special case in which each connected component of $Y$74 is a cycle (Karpinski et al., 6 Sep 2025). In that situation, the relators in $Y$75 are embedded cycles corresponding to relator words, and coning them off means turning each such cycle into a complete graph on its vertices. Equivalently, $Y$76 consists of all group elements represented by subwords of cyclic conjugates of defining relators, and

$Y$77

This makes the clique-cone model especially concrete in the classical setting (Karpinski et al., 6 Sep 2025).

Extreme fineness is automatic in this case because each edge lies in a unique simple closed path; accordingly, one has $Y$78 (Karpinski et al., 6 Sep 2025). The main theorem therefore applies directly to infinitely presented classical $Y$79 groups with finite generating set and finite relators. The paper explicitly includes “classical small cancellation groups” among the groups covered by the result (Karpinski et al., 6 Sep 2025).

This specialization clarifies the geometric meaning of the construction. In the ordinary Cayley graph, a long traversal around a relator cycle has large $Y$80-length. In the coned-off graph, that same traversal has uniformly bounded cost because the cycle has been replaced by a clique. The paper’s general graphical framework extends this intuition from cycles to arbitrary finite connected labeled graphs satisfying graphical small cancellation (Karpinski et al., 6 Sep 2025).

7. Conceptual significance and relation to other cone-off constructions

Within the broader literature on coned-off Cayley graphs, the construction in this work is notable for being both presentation-sensitive and genuinely Cayley-theoretic. The graph $Y$81 is still a Cayley graph of the same group, but for the enlarged generating set $Y$82, and $Y$83 is canonically determined by the path-labels in the chosen defining graph $Y$84 (Karpinski et al., 6 Sep 2025). The paper does not vary the presentation, so the action studied is always the one arising from the specified graphical presentation.

The contrast with relative hyperbolicity is explicit. Standard coned-off graphs for relatively hyperbolic groups add cone-vertices for peripheral cosets; here, one instead replaces each embedded relator by a complete subgraph on its vertex set (Karpinski et al., 6 Sep 2025). What is general is the strategy of collapsing large non-hyperbolic pieces. What is specific in this context is the small-cancellation control over relator intersections, the geodesic decomposition by relators and isolated non-relator edges, the existence and compactness of the bundles $Y$85, and the extreme-fineness argument bounding the number of coding classes in each orbit (Karpinski et al., 6 Sep 2025).

The paper identifies four roles of the coned-off Cayley graph in the proof strategy: it supplies the hyperbolic space whose boundary is studied; its metric defines the correct notion of progress to infinity for rays in the ordinary Cayley graph; its geodesic decomposition by relators creates the combinatorial framework for coding boundary points; and its relator-by-relator structure enables the finite-entry-point argument under extreme fineness (Karpinski et al., 6 Sep 2025). Taken together, these roles show that coned-off Cayley graphs in graphical small cancellation are not merely technical devices for importing hyperbolic methods. They are the primary geometric models through which the boundary, the dynamics, and the hyperfiniteness theorem are simultaneously formulated.