Morse Subsets with Narrow Points

• Morse subsets with narrow points are closed, unbounded subsets in CAT(0) spaces where quasi-geodesics remain uniformly close through a controlled recurrence condition.
• They are characterized by the equivalence of contraction, recurrence, and strong contracting properties, unifying hyperbolic-like behavior in these spaces.
• The narrow point phenomenon imposes a bottleneck effect, forcing detours between distant points to reapproach the set’s interior with explicit quantitative bounds.

Searching arXiv for the cited papers to ground the article in current bibliographic data. arXiv search: (Cashen, 2018) In CAT(0) geometry, a Morse subset is a closed subset $Z$ for which quasi-geodesics with endpoints on $Z$ remain uniformly close to $Z$. For closed, unbounded subsets of complete CAT(0) spaces, Christopher H. Cashen proved that this condition is equivalent to being contracting, recurrent, and strongly contracting, thereby extending earlier results for Morse quasi-geodesics to arbitrary subsets (Cashen, 2018). In the terminology developed around later sublinear boundary theory, the same geometric behavior can be read as a narrow point phenomenon: sufficiently controlled detours between distant parts of $Z$ must return near the interior of $Z$, rather than bypassing it near the endpoints or through higher-rank regions (Qing et al., 2019).

1. Definitions in CAT(0) geometry

A CAT(0) space is a geodesic metric space in which geodesic triangles are slimmer than their Euclidean comparison triangles. Within such a space $X$, a closed subset $Z$ is Morse if there exists a function

$\mu: [1,\infty)\times [0,\infty)\to [0,\infty)$

such that for every $(L,A)$-quasi-geodesic segment $\gamma$ with endpoints on $Z$0, every point $Z$1 satisfies

$Z$2

Thus, quasi-geodesic stability is encoded by a uniform control function depending only on the quasi-geodesic constants, not on the specific endpoints or ambient position (Cashen, 2018).

The complementary contraction language is phrased via the closest-point projection

$Z$3

The associated contraction gauge is

$Z$4

A subset is strongly contracting when $Z$5 is bounded independently of $Z$6, equivalently when

$Z$7

It is contracting when $Z$8. These definitions isolate the hyperbolic-like feature that metric information transverse to $Z$9 collapses under projection.

Cashen also uses a recurrence characterization. For $Z$0,

$Z$1

where $Z$2 ranges over rectifiable segments with endpoints $Z$3, with detour ratio

$Z$4

and where $Z$5 is obtained from $Z$6 by removing the open balls of radius $Z$7 centered at the endpoints. This formulation suppresses endpoint effects and detects whether a controlled detour must return near the middle of $Z$8.

2. Equivalence of Morse, contracting, recurrence, and strong contraction

For closed, unbounded subsets $Z$9 of a complete CAT(0) space $Z$0, the following conditions are equivalent: $Z$1 is Morse; $Z$2 is contracting; $Z$3 is recurrent; and $Z$4 is strongly contracting (Cashen, 2018). The principal new implication established there is the passage from Morse, via recurrence, to strong contraction.

This equivalence is structurally important because it unifies several ways of expressing rank-one or hyperbolic-like behavior. The Morse condition is formulated in terms of stability of quasi-geodesics; contraction is formulated via projections of transverse balls; recurrence is formulated through detour control. The theorem shows that, in the CAT(0) setting and for closed unbounded subsets, these are not merely analogous properties but interchangeable ones.

A common misunderstanding is to regard strong contraction as a phenomenon attached only to quasi-geodesics or geodesic lines. The CAT(0) result shows that this is too restrictive: arbitrary closed Morse subsets exhibit the same bounded projection behavior. This broadens the class of objects to which rank-one methods apply.

3. Recurrence and the narrow point phenomenon

In the material surrounding these results, narrow points refer to a bottleneck property for strongly contracting or Morse sets: every path between far-apart points of $Z$5 must return near a fixed region of $Z$6, rather than avoiding the interior. The recurrence function $Z$7 formalizes exactly this constraint by requiring a point of a controlled detour $Z$8 to lie close to

$Z$9

Because the endpoint balls are excised, the relevant return point is forced away from the ends and into the interior region of the subset (Cashen, 2018).

This exclusion mechanism is the core reason that recurrence is naturally interpreted as narrowness. The statement is not merely that a detour comes back near $Z$0; it is that it comes back near the part of $Z$1 lying between large endpoint neighborhoods. In that sense, the recurrence condition encodes an interior bottleneck.

Strong contraction implies the same qualitative phenomenon. If projection diameters of disjoint balls are uniformly bounded, then long bypasses cannot remain uniformly far from the interior of $Z$2 while still connecting distant points of $Z$3 with controlled detour. The paper’s argument makes this implication quantitative, and the narrow point interpretation is a direct geometric reading of that quantification.

A plausible implication is that narrowness is best understood not as a local thickness condition on $Z$4, but as a global dynamical constraint on how quasi-geodesics and rectifiable detours are allowed to traverse the ambient CAT(0) space.

4. The recurrence-to-strong-contraction argument

The technical core is the proposition that if $Z$5 is a closed, $Z$6-recurrent subset of a CAT(0) space, then $Z$7 is $Z$8-strongly contracting (Cashen, 2018). The proof is explicit and works by contradiction.

Set $Z$9. Assuming $X$0 is not $X$1-strongly contracting, there exist $X$2 with

$X$3

and

$X$4

Choose $X$5 and $X$6 so that

$X$7

Then define $X$8 by removing from $X$9 the open balls of radius $Z$0 centered at $Z$1 and $Z$2.

The argument next constructs, through a case analysis involving the relative positions of $Z$3, a broken geodesic $Z$4 joining points of $Z$5 with

$Z$6

Since $Z$7, recurrence produces a point $Z$8 within distance $Z$9 of $\mu: [1,\infty)\times [0,\infty)\to [0,\infty)$0. The remainder of the proof shows that such a point cannot exist: CAT(0) convexity and triangle inequalities force contradictory bounds such as $\mu: [1,\infty)\times [0,\infty)\to [0,\infty)$1, incompatible with $\mu: [1,\infty)\times [0,\infty)\to [0,\infty)$2. Hence the assumption fails, and the contraction gauge is uniformly bounded by $\mu: [1,\infty)\times [0,\infty)\to [0,\infty)$3.

The significance of the proof lies in its explicitness. The constants are not merely existential; the strong contraction bound is directly expressed in terms of the recurrence function. This makes the relationship among Morse bounds, recurrence data, and contraction data quantitatively transparent.

5. Relation to Sultan’s theorem and the scope of the generalization

Cashen’s theorem generalizes a result of Sultan, who proved that Morse quasi-geodesics in CAT(0) spaces are strongly contracting. Sultan’s proof used asymptotic cones and applied to the image of a quasi-geodesic, relying on a well-defined notion of betweenness coming from the linear order on the quasi-geodesic. Cashen’s argument avoids asymptotic cones and instead uses the recurrence characterization, which applies to arbitrary subsets lacking any intrinsic linear order (Cashen, 2018).

The generalization has two distinct aspects. First, it enlarges the domain from quasi-geodesic images to all closed Morse subsets. Second, it replaces an asymptotic-cone method with a direct geometric proof in the original space. The paper therefore does not merely reprove a known fact with different language; it changes both the level of generality and the proof technology.

This shift also clarifies the role of narrow points. For quasi-geodesics, betweenness can be read directly from the parameter line. For arbitrary subsets, recurrence supplies an alternative interior notion by deleting endpoint balls and demanding return near the remaining part of the subset. In that sense, the recurrence formalism substitutes for linear order.

A common misconception is that the absence of a canonical ordering on a subset prevents a strong bottleneck theorem of the type known for quasi-geodesics. The recurrence-based proof demonstrates that CAT(0) geometry supplies enough structure to recover such a theorem without linear order.

6. Sublinear extensions and narrowness at infinity

The later theory of sublinearly Morse boundaries extends the Morse/contracting correspondence from uniform bounds to sublinear ones. A sublinear function

$\mu: [1,\infty)\times [0,\infty)\to [0,\infty)$4

is monotone increasing, concave, and satisfies

$\mu: [1,\infty)\times [0,\infty)\to [0,\infty)$5

A geodesic ray $\mu: [1,\infty)\times [0,\infty)\to [0,\infty)$6 starting at a basepoint $\mu: [1,\infty)\times [0,\infty)\to [0,\infty)$7 is $\mu: [1,\infty)\times [0,\infty)\to [0,\infty)$8-Morse if, for every $\mu: [1,\infty)\times [0,\infty)\to [0,\infty)$9-quasi-geodesic segment $(L,A)$0 with endpoints on $(L,A)$1, every point $(L,A)$2 of $(L,A)$3 satisfies

$(L,A)$4

where $(L,A)$5. Equivalently, $(L,A)$6 is $(L,A)$7-contracting if there exists $(L,A)$8 such that, for every ball $(L,A)$9 centered at $\gamma$0 and disjoint from $\gamma$1,

$\gamma$2

Theorem A states that a geodesic ray is $\gamma$3-Morse if and only if it is $\gamma$4-contracting (Qing et al., 2019).

In this framework, narrowness becomes asymptotic and scale-sensitive. The paper relates Morse behavior to narrowness at infinity: geodesic rays are forced to stay in tight corridors, and any detour is controlled by a sublinear function. In CAT(0) cube complexes associated to right-angled Artin groups, a geodesic is sublinearly Morse if and only if its projection to every maximal join subcomplex grows at most sublinearly with respect to norm; equivalently, it makes only sublinear-length excursions into join subcomplexes. The stated interpretation is that such rays avoid wandering deeply into higher-rank flats and instead remain in narrow tree-like regions (Qing et al., 2019).

The boundary object

$\gamma$5

is the space of $\gamma$6-equivalence classes of $\gamma$7-contracting geodesic rays starting at the basepoint. It is quasi-isometry invariant and metrizable for proper CAT(0) spaces, and when $\gamma$8 it reduces to the Morse boundary of Charney–Sultan. If $\gamma$9 in the sense that $Z$00, then

$Z$01

with the subspace topology. For Gromov hyperbolic spaces, all such sublinear Morse boundaries coincide with the classical Gromov boundary (Qing et al., 2019).

The right-angled Artin group examples show how narrowness at infinity can be larger than classical Morse behavior while remaining quasi-isometry invariant. For finitely supported random walks on an irreducible right-angled Artin group, almost every sample path tracks a CAT(0) geodesic that is $Z$02-Morse for

$Z$03

The authors state that the corresponding boundary provides a quasi-isometry invariant topological model for the Poisson boundary. This suggests that “narrow points” need not mean uniformly contracting points only; sublinear narrowness can capture asymptotic phenomena that the ordinary Morse boundary is too small to detect.