Sublinearly Morse Bi-infinite Geodesic Line

• Sublinearly Morse bi-infinite geodesic lines are defined by their stability under quasi-geodesic detours, characterized by sublinear contraction in the projection framework.
• The projection-theoretic approach demonstrates that points distant from the line project into sublinearly shrinking neighborhoods, revealing hyperbolic-like behavior in broader metric spaces.
• Divergence analysis establishes that these geodesic lines exhibit completely superlinear divergence, with significant implications in CAT(0) geometry, hierarchical spaces, and stochastic models.

A sublinearly Morse bi-infinite geodesic line is, in the projection-theoretic framework of Arzhantseva–Cashen–Gruber–Hume, a bi-infinite geodesic line $\ell\subset X$ in a geodesic metric space whose closest-point projection is sublinearly contracting; equivalently, $\ell$ is Morse; equivalently, viewed as a quasi-geodesic, it has completely superlinear divergence (Arzhantseva et al., 2016). The notion isolates a two-ended form of “hyperbolic-like” behavior in spaces that need not be globally hyperbolic, and later work reinterprets the same large-scale phenomenon through $\kappa$-Morse stability, boundary constructions, combinatorial criteria, and probabilistic applications.

1. Foundational definition

Let $X$ be a geodesic metric space and let $\ell=\gamma(\mathbb R)\subset X$ be the image of an isometric embedding $\gamma:\mathbb R\to X$. In the sense of (Arzhantseva et al., 2016), the relevant starting point is the Morse property for a subspace $Y\subset X$: $Y$ is $\mu$-Morse if, for every $L\ge 1$ and $\ell$0, every $\ell$1-quasi-geodesic with endpoints on $\ell$2 lies in $\ell$3. When such a function $\ell$4 exists, $\ell$5 is called Morse (Arzhantseva et al., 2016).

Specialized to a bi-infinite geodesic line, this means that every quasi-geodesic in $\ell$6 with endpoints on $\ell$7 stays in a uniformly controlled neighborhood of $\ell$8, with control depending only on the quasi-geodesic constants. In that sense, the line is stable under quasi-geodesic detours. The same paper emphasizes an equivalent viewpoint for quasi-geodesics: a quasi-geodesic is Morse if and only if the collection of all its subsegments is uniformly Morse (Arzhantseva et al., 2016).

This formulation is already two-sided. A bi-infinite line is not treated merely as an unparameterized subset but also as a quasi-geodesic $\ell$9, so both the subspace structure and the parameterized geodesic structure are available simultaneously. That dual viewpoint underlies all later characterizations.

2. Projection-theoretic characterization

The projection-theoretic language in (Arzhantseva et al., 2016) uses the $\kappa$0-closest point projection to a subspace $\kappa$1: $\kappa$2 The projection $\kappa$3 is $\kappa$4-contracting when $\kappa$5 are non-decreasing and eventually non-negative, $\kappa$6 is unbounded and satisfies $\kappa$7, and whenever

$\kappa$8

one has

$\kappa$9

together with

$X$0

The key special case is $X$1. Then $X$2 is called sublinearly contracting, and $X$3 is itself sublinear: $X$4 (Arzhantseva et al., 2016).

For a bi-infinite geodesic line $X$5, saying that $X$6 is sublinearly Morse means exactly that $X$7 is $X$8-contracting for some sublinear $X$9. The main equivalence theorem for subspaces states that, for a subspace $\ell=\gamma(\mathbb R)\subset X$0,

$\ell=\gamma(\mathbb R)\subset X$1

(Arzhantseva et al., 2016). For lines, this is not a separate theorem but a direct specialization.

The same paper also gives a geodesic-segment characterization. For a subspace $\ell=\gamma(\mathbb R)\subset X$2, the following are equivalent: there exist sublinear functions controlling the diameter of $\ell=\gamma(\mathbb R)\subset X$3 for geodesic segments $\ell=\gamma(\mathbb R)\subset X$4 that stay sufficiently far from $\ell=\gamma(\mathbb R)\subset X$5; and $\ell=\gamma(\mathbb R)\subset X$6 is $\ell=\gamma(\mathbb R)\subset X$7-contracting for some sublinear $\ell=\gamma(\mathbb R)\subset X$8 (Arzhantseva et al., 2016). For a line $\ell=\gamma(\mathbb R)\subset X$9, the geometric content is that if a geodesic segment stays far from $\gamma:\mathbb R\to X$0, then its projection to $\gamma:\mathbb R\to X$1 has small diameter, sublinearly small in the relevant distance scale. This is the projection-theoretic expression of the statement that far-away points project to sets whose diameters are tiny compared with the distance from the line.

3. Divergence and detour geometry

A second characterization in (Arzhantseva et al., 2016) is divergence. For an $\gamma:\mathbb R\to X$2-quasi-geodesic $\gamma:\mathbb R\to X$3, parameters $\gamma:\mathbb R\to X$4, $\gamma:\mathbb R\to X$5, and scales $\gamma:\mathbb R\to X$6, the quantity

$\gamma:\mathbb R\to X$7

is the infimal length of a path from $\gamma:\mathbb R\to X$8 to $\gamma:\mathbb R\to X$9 that avoids the ball centered at $Y\subset X$0 of radius

$Y\subset X$1

The divergence is then

$Y\subset X$2

Up to the paper’s coarse equivalence relation $Y\subset X$3, this gives a well-defined divergence class $Y\subset X$4 (Arzhantseva et al., 2016).

The refined notion is completely superlinear divergence. A function $Y\subset X$5 is completely super-$Y\subset X$6 if for every $Y\subset X$7 and $Y\subset X$8, the set of $Y\subset X$9 such that

$Y$0

is bounded. Taking $Y$1 yields completely superlinear divergence. The main theorem for quasi-geodesics is

$Y$2

(Arzhantseva et al., 2016). For a bi-infinite geodesic line, this is one of the standard equivalent characterizations of sublinearly Morse behavior.

This detour geometry has a concrete CAT(0) realization. In Davis complexes of right-angled Coxeter groups, there are Morse bi-infinite geodesics with divergence equivalent to $Y$3 for every real $Y$4; for each integer $Y$5, there is a CAT(0) space $Y$6 containing Morse geodesics with divergence equivalent to $Y$7 for every $Y$8 (Tran, 2014). The same source states that a bi-infinite geodesic in a CAT(0) space is Morse if and only if it has superlinear divergence, and its constructions answer the Behrstock–Druţu question by producing examples with divergence strictly between consecutive integer powers (Tran, 2014). These examples show that the divergence side of the theory is not limited to exponential or quadratic growth.

4. Later $Y$9-Morse formulations and the endwise interpretation

Later boundary-oriented work fixes a concave sublinear function

$\mu$0

and defines a $\mu$1-neighborhood by

$\mu$2

or, with basepoint notation,

$\mu$3

(Qing et al., 2020, Jana et al., 10 Jul 2025). In this framework, a closed set $\mu$4 is $\mu$5-Morse if quasi-geodesics with endpoints on $\mu$6 remain in a $\mu$7-controlled neighborhood of $\mu$8, and there is also a ray-based stability formulation in which quasi-geodesic rays that come sublinearly close far out must fellow-travel long initial segments with sublinear error control (Qing et al., 2020, Jana et al., 10 Jul 2025).

For a bi-infinite geodesic line $\mu$9, the natural specialization in this literature is explicitly endwise: each of the two rays $L\ge 1$0 and $L\ge 1$1 must be $L\ge 1$2-Morse with compatible control, so that the line defines two boundary points, one for each end rather than a single point (Nguyen et al., 2022). The formal boundaries are built from rays, not lines, but the same sublinear stability estimates are the ones used to treat lines endwise (Nguyen et al., 2022, Qing et al., 2020).

This later usage is weaker than the classical Morse condition. A standard Morse geodesic uses a uniformly bounded neighborhood, whereas here the neighborhood is allowed to grow like

$L\ge 1$3

which is sublinear in the distance to the basepoint (Jana et al., 10 Jul 2025). This suggests that the phrase “sublinearly Morse” is used in two related senses in the literature: in (Arzhantseva et al., 2016), sublinear contraction is an equivalent characterization of ordinary Morse behavior for lines, while in later $L\ge 1$4-boundary work it denotes stability up to sublinear tubes.

A further characterization of the later notion is middle recurrence. For a bi-infinite geodesic line $L\ge 1$5, the middle third of $L\ge 1$6 is

$L\ge 1$7

The line is $L\ge 1$8-middle recurrent if every path $L\ge 1$9 with endpoints $\ell$00 and

$\ell$01

meets a sublinear neighborhood of the middle third: $\ell$02 The 2026 characterization states that a quasi-geodesic ray is sublinearly Morse if and only if it is $\ell$03-middle recurrent for some sublinear $\ell$04 (Jana et al., 25 Mar 2026). In this form, the line cannot be bypassed cheaply in the middle.

5. CAT(0), cubical, and hierarchical characterizations

In proper CAT(0) spaces, the later literature uses the known equivalence

$\ell$05

for geodesic rays (Vest, 2023). One combinatorial realization is the curtain machinery of Petyt–Spriano–Zalloum. Given a geodesic $\ell$06 and a parameter $\ell$07, the curtain dual to $\ell$08 at $\ell$09 is

$\ell$10

A ray is $\ell$11-contracting if and only if it is a $\ell$12-curtain-excursion geodesic, meaning that it admits a chain of curtains whose spacing and separation are controlled by $\ell$13 (Vest, 2023). For a bi-infinite line, the same interpretation applies to each end: both ends admit controlled curtain-excursion structures, and under suitable hypotheses the sublinearly Morse boundary injects continuously into the Gromov boundary of the hyperbolic curtain model $\ell$14 (Vest, 2023).

In finite-dimensional CAT(0) cube complexes, the corresponding combinatorial criterion is hyperplane-theoretic. A geodesic ray $\ell$15 is $\ell$16-contracting if and only if there exists an infinite sequence of hyperplanes $\ell$17 crossed at points $\ell$18 such that

$\ell$19

and

$\ell$20

(Incerti-Medici et al., 2021). The same paper is explicit that its primary theory is ray-based rather than line-based, but it also proves that every geodesic line crosses a bi-infinite chain of hyperplanes (Incerti-Medici et al., 2021). That supplies the combinatorial prerequisite for a two-ended interpretation.

For hierarchically hyperbolic spaces, mapping class groups, and Teichmüller space, the main characterization is via persistent shadow and bounded projections. In a proper HHS with unbounded products, there exists $\ell$21 such that $\ell$22-Morse rays have $\ell$23-persistent shadow, and median rays with $\ell$24-persistent shadow or $\ell$25-bounded projections are $\ell$26-Morse after the appropriate power correction (Durham et al., 2022). The same source proves that the sublinearly Morse boundary is a visibility space: any two distinct points in $\ell$27 are joined by a bi-infinite $\ell$28-Morse geodesic line (Durham et al., 2022). In CAT(0) admissible groups, the quantitative specialization is

$\ell$29

where $\ell$30 is the HHS complexity, and the $\ell$31-Morse boundary is used as a topological model for associated Poisson boundaries (Nguyen et al., 2022).

6. Boundary theory, examples, and stochastic robustness

The boundary point of view is built from rays modulo sublinear fellow traveling: $\ell$32 For proper geodesic metric spaces, the $\ell$33-Morse boundary is quasi-isometrically invariant and metrizable (Qing et al., 2020). Under suitable sublinear biLipschitz equivalences, it is also invariant under SBE: if $\ell$34 is a $\ell$35-SBE and $\ell$36 dominates $\ell$37, then

$\ell$38

is a homeomorphism (Pallier et al., 2022). These results make the boundary data associated to the two ends of a sublinearly Morse line stable under large-scale deformations.

The dynamical theory is likewise formulated on boundaries of rays but has direct implications for bi-infinite axes. If $\ell$39 is cobounded and $\ell$40, then every $\ell$41-orbit is dense in $\ell$42; moreover, contracting elements or sublinearly Morse isometries induce weak north-south dynamics on $\ell$43 (Garcia et al., 2024). In proper cocompact CAT(0) spaces, the quasi-redirecting boundary is a visibility space, so there exists a bi-infinite geodesic line connecting every pair of directions (Garcia et al., 2024).

Existence results show that such lines occur far beyond periodic axes of infinite-order elements. In graded small cancellation torsion groups $\ell$44 with $\ell$45, there is an isometrically embedded $\ell$46-regular tree $\ell$47 in which every bi-infinite simple path is a Morse geodesic (Fink, 2017). The significance is that the group has Morse geodesics but no Morse elements, since the torsion hypothesis rules out infinite-order elements (Fink, 2017).

Probabilistic work shows that sublinear Morse geometry is robust under first passage percolation. If an infinite connected bounded-degree graph contains a Morse quasi-geodesic, then under the assumptions

$\ell$48

the FPP metric almost surely admits a bi-infinite geodesic (Benjamini et al., 2016). The sublinear analogue replaces the Morse hypothesis by the existence of a sublinearly Morse bi-infinite quasi-geodesic line and again concludes that almost surely the FPP graph contains a bi-infinite geodesic line (Jana et al., 10 Jul 2025). A further 2026 result proves that, under the same bounded-degree and edge-weight hypotheses, sublinearly Morse boundaries are almost surely preserved under FPP: $\ell$49 by a homeomorphism induced by FPP (Jana et al., 25 Mar 2026). In that setting, the middle recurrence characterization is the key mechanism behind preservation.

Taken together, these developments place the sublinearly Morse bi-infinite geodesic line at the intersection of projection geometry, divergence theory, combinatorics of curtains and hyperplanes, boundary dynamics, and stochastic geometry. In the original projection-theoretic sense of (Arzhantseva et al., 2016), such a line is exactly a Morse line characterized by sublinear contraction and completely superlinear divergence. In the later $\ell$50-Morse sense, it is a line whose two ends remain stable under sublinear rather than bounded error. Both viewpoints formalize the same guiding idea: efficient detours between points on the line cannot escape its large-scale influence except by paying a cost that is visible at sublinear scale.