---
title: Sublinearly Morse Bi-infinite Geodesic Line
url: https://www.emergentmind.com/topics/sublinearly-morse-bi-infinite-geodesic-line
type: topic
---

# Sublinearly Morse Bi-infinite Geodesic Line

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 [1601.01897]. 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 [1601.01897], 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 \(A\ge 0\), every \((L,A)\)-quasi-geodesic with endpoints on \(Y\) lies in \(N_{\mu(L,A)}(Y)\). When such a function \(\mu\) exists, \(Y\) is called Morse [1601.01897].

Specialized to a bi-infinite geodesic line, this means that every quasi-geodesic in \(X\) with endpoints on \(\ell\) stays in a uniformly controlled neighborhood of \(\ell\), 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 [1601.01897].

This formulation is already two-sided. A bi-infinite line is not treated merely as an unparameterized subset but also as a quasi-geodesic \(\gamma:\mathbb R\to X\), 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 [1601.01897] uses the \(\eta\)-closest point projection to a subspace \(Y\subset X\):
\[
\pi_Y^\eta(x)=\{\,y\in Y\mid d(x,y)\le d(x,Y)+\eta\,\}.
\]
The projection \(\pi_Y^\eta\) is \((\rho_1,\rho_2)\)-contracting when \(\rho_1,\rho_2\) are non-decreasing and eventually non-negative, \(\rho_1\) is unbounded and satisfies \(\rho_1(r)\le r\), and whenever
\[
d(x,x')\le \rho_1(d(x,Y)),
\]
one has
\[
\operatorname{diam}\bigl(\pi_Y^\eta(x)\cup \pi_Y^\eta(x')\bigr)\le \rho_2(d(x,Y)),
\]
together with
\[
\lim_{r\to\infty}\frac{\rho_2(r)}{\rho_1(r)}=0.
\]
The key special case is \(\rho_1(r)=r\). Then \(\pi_Y^\eta\) is called sublinearly contracting, and \(\rho_2\) is itself sublinear:
\[
\lim_{r\to\infty}\frac{\rho_2(r)}{r}=0
\]
[1601.01897].

For a bi-infinite geodesic line \(\ell\), saying that \(\ell\) is sublinearly Morse means exactly that \(\pi_\ell^\eta\) is \((r,\rho)\)-contracting for some sublinear \(\rho\). The main equivalence theorem for subspaces states that, for a subspace \(Y\subset X\),
\[
Y \text{ is Morse} \iff Y \text{ is sublinearly contracting} \iff Y \text{ is }(\rho_1,\rho_2)\text{-contracting for some }\rho_1,\rho_2
\]
[1601.01897]. For lines, this is not a separate theorem but a direct specialization.

The same paper also gives a geodesic-segment characterization. For a subspace \(Y\), the following are equivalent: there exist sublinear functions controlling the diameter of \(\pi_Y^\eta(\gamma)\) for geodesic segments \(\gamma\) that stay sufficiently far from \(Y\); and \(\pi_Y^\eta\) is \((r,\rho)\)-contracting for some sublinear \(\rho\) [1601.01897]. For a line \(\ell\), the geometric content is that if a geodesic segment stays far from \(\ell\), then its projection to \(\ell\) 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 [1601.01897] is divergence. For an \((L,A)\)-quasi-geodesic \(\gamma:\mathbb R\to X\), parameters \(\lambda\in(0,1]\), \(\kappa\ge L+A\), and scales \(r,s\), the quantity
\[
\Lambda_\gamma(r,s;L,A,\lambda,\kappa)
\]
is the infimal length of a path from \(\gamma(s-r)\) to \(\gamma(s+r)\) that avoids the ball centered at \(\gamma(s)\) of radius
\[
\lambda(L^{-1}r-A)-\kappa.
\]
The divergence is then
\[
\Delta_\gamma(r;L,A,\lambda,\kappa)=\inf_s \Lambda_\gamma(r,s;L,A,\lambda,\kappa).
\]
Up to the paper’s coarse equivalence relation \(\asymp\), this gives a well-defined divergence class \(\Delta_\gamma(r)\) [1601.01897].

The refined notion is completely superlinear divergence. A function \(g\) is completely super-\(f\) if for every \(C_1,C_2>0\) and \(C_3,C_4\ge 0\), the set of \(r\) such that
\[
g(r)\le C_1 f(C_2r+C_3)+C_4
\]
is bounded. Taking \(f(r)=r\) yields completely superlinear divergence. The main theorem for quasi-geodesics is
\[
\gamma \text{ is Morse} \iff \gamma \text{ has completely superlinear divergence}
\]
[1601.01897]. 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 \(r^s\) for every real \(s\ge 2\); for each integer \(m\ge 2\), there is a CAT(0) space \(Y_m\) containing Morse geodesics with divergence equivalent to \(r^s\) for every \(s\in[2,m]\) [1408.6089]. 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 [1408.6089]. These examples show that the divergence side of the theory is not limited to exponential or quadratic growth.

## 4. Later \(\kappa\)-Morse formulations and the endwise interpretation

Later boundary-oriented work fixes a concave sublinear function
\[
\kappa:[0,\infty)\to[1,\infty), \qquad \lim_{t\to\infty}\frac{\kappa(t)}{t}=0,
\]
and defines a \(\kappa\)-neighborhood by
\[
N_\kappa(Z,c)=\{x\in X:\ d(x,Z)\le c\,\kappa(x)\}
\]
or, with basepoint notation,
\[
N_\kappa(Z,c)=\{x\in X : d_X(x,Z)\le c\,\kappa(\|x\|)\}
\]
[2011.03481; 2507.07859]. In this framework, a closed set \(Z\) is \(\kappa\)-Morse if quasi-geodesics with endpoints on \(Z\) remain in a \(\kappa\)-controlled neighborhood of \(Z\), 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 [2011.03481; 2507.07859].

For a bi-infinite geodesic line \(\gamma:\mathbb R\to X\), the natural specialization in this literature is explicitly endwise: each of the two rays \(\gamma|_{[0,\infty)}\) and \(\gamma|_{(-\infty,0]}\) must be \(\kappa\)-Morse with compatible control, so that the line defines two boundary points, one for each end rather than a single point [2203.00935]. The formal boundaries are built from rays, not lines, but the same sublinear stability estimates are the ones used to treat lines endwise [2203.00935; 2011.03481].

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
\[
d(x,Z)\lesssim C\,\kappa(\|x\|),
\]
which is sublinear in the distance to the basepoint [2507.07859]. This suggests that the phrase “sublinearly Morse” is used in two related senses in the literature: in [1601.01897], sublinear contraction is an equivalent characterization of ordinary Morse behavior for lines, while in later \(\kappa\)-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 \(\gamma\), the middle third of \(\gamma[a,b]\) is
\[
\gamma_{\frac13[a,b]}:=\left\{x\in\gamma:\ \min\{d(x,a),d(x,b)\}\ge \tfrac13\,d(a,b)\right\}.
\]
The line is \(\kappa\)-middle recurrent if every path \(P\) with endpoints \(a,b\in\gamma\) and
\[
\ell(P)\le C\,d(a,b)
\]
meets a sublinear neighborhood of the middle third:
\[
P\cap N_\kappa\!\left(\gamma_{\frac13[a,b]},c\right)\neq\emptyset.
\]
The 2026 characterization states that a quasi-geodesic ray is sublinearly Morse if and only if it is \(\kappa\)-middle recurrent for some sublinear \(\kappa\) [2603.24423]. 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
\[
\text{\(\kappa\)-contracting} \iff \text{\(\kappa\)-Morse}
\]
for geodesic rays [2309.02725]. One combinatorial realization is the curtain machinery of Petyt–Spriano–Zalloum. Given a geodesic \(b:I\to X\) and a parameter \(r\), the curtain dual to \(b\) at \(r\) is
\[
h_{b,r}=\pi_b^{-1}\big(b[r-\tfrac12,r+\tfrac12]\big).
\]
A ray is \(\kappa\)-contracting if and only if it is a \(\kappa\)-curtain-excursion geodesic, meaning that it admits a chain of curtains whose spacing and separation are controlled by \(\kappa\) [2309.02725]. 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 \(\widehat X\) [2309.02725].

In finite-dimensional CAT(0) cube complexes, the corresponding combinatorial criterion is hyperplane-theoretic. A geodesic ray \(b\) is \(\kappa\)-contracting if and only if there exists an infinite sequence of hyperplanes \(h_1,h_2,\dots\) crossed at points \(b(t_i)\) such that
\[
d(t_i,t_{i+1})\le \kappa(t_{i+1})
\]
and
\[
h_i,h_{i+1}\text{ are }\kappa(t_{i+1})\text{-well-separated}
\]
[2101.01037]. 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 [2101.01037]. 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 \(p=p(X)>0\) such that \(\kappa\)-Morse rays have \(\kappa^p\)-persistent shadow, and median rays with \(\kappa\)-persistent shadow or \(\kappa\)-bounded projections are \(\kappa\)-Morse after the appropriate power correction [2207.06516]. The same source proves that the sublinearly Morse boundary is a visibility space: any two distinct points in \(\partial_\kappa X\) are joined by a bi-infinite \(\kappa\)-Morse geodesic line [2207.06516]. In CAT(0) admissible groups, the quantitative specialization is
\[
\kappa(r)=\log^p(r),
\]
where \(p\) is the HHS complexity, and the \(\kappa\)-Morse boundary is used as a topological model for associated Poisson boundaries [2203.00935].

## 6. Boundary theory, examples, and stochastic robustness

The boundary point of view is built from rays modulo sublinear fellow traveling:
\[
\beta\sim\alpha \Longleftrightarrow \lim_{r\to\infty}\frac{d_X(\alpha_r,\beta_r)}{r}=0.
\]
For proper geodesic metric spaces, the \(\kappa\)-Morse boundary is quasi-isometrically invariant and metrizable [2011.03481]. Under suitable sublinear biLipschitz equivalences, it is also invariant under SBE: if \(\Phi:X\to Y\) is a \((L,\theta)\)-SBE and \(\kappa\) dominates \(\theta\), then
\[
\Phi_\star:\partial_\kappa X\to\partial_\kappa Y
\]
is a homeomorphism [2211.01023]. 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 \(G\curvearrowright X\) is cobounded and \(|\partial_K X|>3\), then every \(G\)-orbit is dense in \(\partial_K X\); moreover, contracting elements or sublinearly Morse isometries induce weak north-south dynamics on \(\partial_K X\) [2408.10105]. 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 [2408.10105].

Existence results show that such lines occur far beyond periodic axes of infinite-order elements. In graded small cancellation torsion groups \(G=\lim_{i\to\infty}G_i\) with \(G_0=F_4\), there is an isometrically embedded \(7\)-regular tree \(T\) in which every bi-infinite simple path is a Morse geodesic [1710.11191]. The significance is that the group has Morse geodesics but no Morse elements, since the torsion hypothesis rules out infinite-order elements [1710.11191].

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
\[
\mathbb E[\omega(e)]<\infty,\qquad \nu(\{0\})=0,
\]
the FPP metric almost surely admits a bi-infinite geodesic [1606.02449]. 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 [2507.07859]. A further 2026 result proves that, under the same bounded-degree and edge-weight hypotheses, sublinearly Morse boundaries are almost surely preserved under FPP:
\[
\partial_s X \cong \partial_s X_\omega
\]
by a homeomorphism induced by FPP [2603.24423]. 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 [1601.01897], such a line is exactly a Morse line characterized by sublinear contraction and completely superlinear divergence. In the later \(\kappa\)-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.

Source: https://www.emergentmind.com/topics/sublinearly-morse-bi-infinite-geodesic-line