Proportionally Contracting Rays in Metric Spaces

• Proportionally contracting rays are geodesic rays in metric spaces whose long subsegments satisfy uniform contraction over a fixed asymptotic proportion.
• They relax the requirement for global contraction by focusing on density, thereby extending classical contracting and Morse frameworks.
• Their genericity in non-amenable groups and CAT(0) spaces underpins the application of Ancona inequalities and advances probabilistic boundary theory.

Searching arXiv for the specified paper and closely related work on contracting boundaries and contracting geodesics. Proportionally contracting rays are geodesic rays in a geodesic metric space for which, asymptotically, a prescribed positive proportion of the ray is occupied by long uniformly contracting subsegments. In the formulation introduced for groups with contracting elements, a geodesic ray $\gamma : [0,\infty) \to X$ is called a $\theta$-proportionally $(D,L)$-contracting ray if

$$\liminf_{n \to \infty} \frac{\|\mathrm{contr}_{(D,L)}(\gamma[0, n])\|}{n} \geq \theta,$$

where $\mathrm{contr}_{(D,L)}(\cdot)$ denotes the total length of subsegments of $\gamma[0,n]$ that are $D$-contracting and have length at least $L$ (Liu et al., 14 Sep 2025). The notion was developed in the study of Ancona inequalities along generic geodesic rays in non-amenable groups with contracting elements, and it extends the utility of contracting or Morse geometry beyond the setting in which an entire ray is uniformly Morse or strongly contracting (Liu et al., 14 Sep 2025).

1. Definition and formal framework

The ambient setting is a geodesic metric space $(X,d)$. Fix constants $\theta \in (0,1]$, $\theta$0, and $\theta$1. A geodesic ray $\theta$2 is a $\theta$3-proportionally $\theta$4-contracting ray if the liminf proportion of its initial segment covered by long $\theta$5-contracting subsegments is at least $\theta$6 (Liu et al., 14 Sep 2025).

This definition isolates an asymptotic density condition rather than a uniform condition along the full ray. In particular, it does not require every subsegment of the ray to be contracting, nor does it require a bounded contraction function on the whole ray. Instead, for large $\theta$7, at least a $\theta$8 fraction of the length of $\theta$9 is composed of long $(D,L)$0-contracting subsegments (Liu et al., 14 Sep 2025). This suggests that proportional contraction is a density-based weakening of global contraction, designed to retain enough hyperbolic structure to support boundary and potential-theoretic arguments.

The contracting condition used on subsegments is the standard projection-control condition. A subsegment $(D,L)$1 of a geodesic is $(D,L)$2-contracting if for any points $(D,L)$3 with $(D,L)$4, the diameter of the projections of $(D,L)$5 and $(D,L)$6 to $(D,L)$7 is at most $(D,L)$8 (Liu et al., 14 Sep 2025). In CAT(0) spaces, contracting geodesics admit equivalent descriptions via Morse, slim, and divergence properties; in particular, for a geodesic ray or line, contracting, Morse, slim, superlinear lower divergence, and at least quadratic lower divergence are equivalent (Charney et al., 2013). In proper geodesic metric spaces more generally, Morse and contracting properties are equivalent for the boundary constructions considered in the contracting-boundary literature (Cashen, 2016).

2. Relation to contracting and strongly contracting geometry

Proportionally contracting rays sit within a broader hierarchy of hyperbolic-like geodesic behaviors. A geodesic ray $(D,L)$9 is contracting if there exists a non-decreasing, eventually non-negative function $$\liminf_{n \to \infty} \frac{\|\mathrm{contr}_{(D,L)}(\gamma[0, n])\|}{n} \geq \theta,$$0 with $$\liminf_{n \to \infty} \frac{\|\mathrm{contr}_{(D,L)}(\gamma[0, n])\|}{n} \geq \theta,$$1 such that

$$\liminf_{n \to \infty} \frac{\|\mathrm{contr}_{(D,L)}(\gamma[0, n])\|}{n} \geq \theta,$$2

for all $$\liminf_{n \to \infty} \frac{\|\mathrm{contr}_{(D,L)}(\gamma[0, n])\|}{n} \geq \theta,$$3 (Cashen, 2016). If $$\liminf_{n \to \infty} \frac{\|\mathrm{contr}_{(D,L)}(\gamma[0, n])\|}{n} \geq \theta,$$4 can be chosen bounded, then $$\liminf_{n \to \infty} \frac{\|\mathrm{contr}_{(D,L)}(\gamma[0, n])\|}{n} \geq \theta,$$5 is strongly contracting (Cashen, 2016). In CAT(0) spaces, a geodesic is $$\liminf_{n \to \infty} \frac{\|\mathrm{contr}_{(D,L)}(\gamma[0, n])\|}{n} \geq \theta,$$6-contracting if

$$\liminf_{n \to \infty} \frac{\|\mathrm{contr}_{(D,L)}(\gamma[0, n])\|}{n} \geq \theta,$$7

equivalently every ball disjoint from $$\liminf_{n \to \infty} \frac{\|\mathrm{contr}_{(D,L)}(\gamma[0, n])\|}{n} \geq \theta,$$8 projects to a segment of length $$\liminf_{n \to \infty} \frac{\|\mathrm{contr}_{(D,L)}(\gamma[0, n])\|}{n} \geq \theta,$$9 on $\mathrm{contr}_{(D,L)}(\cdot)$0 (Charney et al., 2013).

The distinction between strong contraction and weaker sublinear contraction is central in the contracting-boundary literature. Explicit examples show rays that are contracting but not strongly contracting, with projection diameter growing logarithmically rather than remaining bounded (Cashen, 2016). In that context, the contrast between strong and proportional contraction is critical: rays with proportionally sublinear contraction functions, such as $\mathrm{contr}_{(D,L)}(\cdot)$1, expose pathologies of the Gromov product topology on the contracting boundary under quasi-isometries (Cashen, 2016). A plausible implication is that the newer notion of proportionally contracting rays shifts attention from uniform control of a single ray to the statistical prevalence of uniformly contracting pieces along a generic ray.

This position between global contraction and unrestricted geodesicity is also reflected by comparison with strongly contracting geodesics in other settings. In Outer space, strong contraction is characterized by bounded diameter of closest-point projections under a metric-ball separation condition, and nondegenerate strongly contracting geodesics project to parameterized quasigeodesics in the free factor complex (Dowdall et al., 2015). In finitely generated groups, the language of $\mathrm{contr}_{(D,L)}(\cdot)$2-super-contracting geodesics in a Cayley graph is regular for every $\mathrm{contr}_{(D,L)}(\cdot)$3 (Eike et al., 2018). Proportionally contracting rays are not presented there, but those results situate the concept within an existing program that treats contraction as a marker of hyperbolic behavior inside non-hyperbolic spaces.

3. Construction, inheritance, and good points

The construction described for groups with contracting elements begins with a geodesic ray $\mathrm{contr}_{(D,L)}(\cdot)$4 in a proper geodesic space admitting a geometric group action by a group with contracting elements. One decomposes $\mathrm{contr}_{(D,L)}(\cdot)$5 into maximal subsegments that are $\mathrm{contr}_{(D,L)}(\cdot)$6-contracting and of length at least $\mathrm{contr}_{(D,L)}(\cdot)$7; if the total length of these segments in each large initial segment is eventually at least a fraction $\mathrm{contr}_{(D,L)}(\cdot)$8 of the whole, then $\mathrm{contr}_{(D,L)}(\cdot)$9 is proportionally contracting (Liu et al., 14 Sep 2025).

Several structural properties are emphasized. First, every proportionally contracting ray contains infinitely many good points, namely locations along the ray from which, for any distant subsegment extending further along the ray, a proportional contraction property holds (Liu et al., 14 Sep 2025). Second, subrays and long subsegments inherit proportional contraction properties, possibly with weaker parameters (Liu et al., 14 Sep 2025). Third, these rays satisfy strong divergence properties ensuring that any path far from the ray must be long compared to the projected distance along the ray (Liu et al., 14 Sep 2025).

The good-point formalism is essential for later applications. If $\gamma[0,n]$0 is a $\gamma[0,n]$1-good point on a geodesic $\gamma[0,n]$2, then multiplicative estimates for the Green function hold for suitable antipodal configurations near $\gamma[0,n]$3 (Liu et al., 14 Sep 2025). This suggests that the combinatorial density of contracting pieces can be localized into an infinite sequence of analytically useful points.

4. Genericity and typicality

A central feature of the theory is that proportionally contracting rays are generic in both counting and measure-theoretic senses. They make up a set of full measure with respect to natural Patterson–Sullivan conformal measures on the horofunction boundary, and for appropriate choices of parameters they are exponentially generic in counting measure among all rays or geodesic elements (Liu et al., 14 Sep 2025).

More concretely, the set of group elements for which the geodesic from a basepoint $\gamma[0,n]$4 to $\gamma[0,n]$5 is proportionally contracting has exponentially large growth, while the complement is exponentially small (Liu et al., 14 Sep 2025). Likewise, the set of boundary points admitting at least one proportionally contracting ray ending at that point has full measure in the horofunction boundary (Liu et al., 14 Sep 2025). The paper summarizes this by stating that such rays arise as endpoints for “most” geodesics and “most” points on the boundary, both in measure and in counting (Liu et al., 14 Sep 2025).

This genericity differentiates the notion from classical Morse-ray theory. Morse geodesics are often geometrically rigid but may form a thin subset. Proportionally contracting rays are designed to capture a full-measure class in non-amenable groups with contracting elements (Liu et al., 14 Sep 2025). A plausible implication is that the concept serves as a bridge between sparse rank-one or Morse phenomena and boundary theories that require almost-everywhere statements.

5. Ancona inequalities along proportionally contracting rays

The main analytical motivation for introducing the notion is the extension of Ancona inequalities beyond globally Morse geodesics. Ancona inequalities assert that the Green function is coarsely multiplicative along suitable sequences of points on a geodesic, a fact that is fundamental in the identification of Martin boundaries (Liu et al., 14 Sep 2025).

For a $\gamma[0,n]$6-good point $\gamma[0,n]$7 on a geodesic $\gamma[0,n]$8, and for any $\gamma[0,n]$9, there is a constant $D$0 such that for any $D$1-antipodal pair of points $D$2 along $D$3 with middle point near $D$4,

$D$5

where $D$6 is the Green function of the random walk (Liu et al., 14 Sep 2025). Because every proportionally contracting ray contains infinitely many good points, such inequalities apply at unboundedly many places along a generic ray (Liu et al., 14 Sep 2025).

The broader context is that the same work proves several versions of the Ancona inequality for finitely supported, irreducible random walks on non-amenable groups. It first studies Morse subsets with narrow points, obtaining Ancona inequalities around these points in any finitely generated non-amenable group, which implies the inequality along all Morse geodesics and recovers the relatively hyperbolic case (Liu et al., 14 Sep 2025). Proportionally contracting rays then extend this framework to a generic class of rays in groups acting geometrically with contracting elements (Liu et al., 14 Sep 2025).

6. Boundary theory and geometric applications

The existence of Ancona inequalities along proportionally contracting rays yields a partial boundary map from a full-measure subset of a geometric boundary to the minimal Martin boundary. In the horofunction setting, there exists a full-measure subset $D$7 and a $D$8-equivariant map

$D$9

that is injective up to the finite-difference equivalence relation, continuous in an appropriate topology, and whose image is minimal (Liu et al., 14 Sep 2025). In CAT(0) spaces and CAT(0) cube complexes where the finite-difference relation is trivial, the map is actually injective (Liu et al., 14 Sep 2025).

A stronger Ancona inequality is obtained for groups acting geometrically on an irreducible CAT(0) cube complex with a Morse hyperplane; in that case orbital maps extend continuously to a partial boundary map from a full-measure subset of the Roller boundary into the minimal Martin boundary (Liu et al., 14 Sep 2025). The same paper provides explicit examples, including right-angled Coxeter groups defined by an irreducible graph with at least one vertex not belonging to any induced $L$0-cycle (Liu et al., 14 Sep 2025).

These applications connect the notion to earlier contracting-boundary constructions. For CAT(0) spaces, the contracting boundary $L$1 consists of asymptotic classes of contracting rays and is quasi-isometry invariant (Charney et al., 2013). Later work introduced the topology of fellow-travelling quasi-geodesics on the contracting boundary, a quasi-isometry invariant topology that is metrizable for finitely generated groups (Cashen et al., 2017). By contrast, the Gromov product topology on the contracting boundary is not quasi-isometry invariant in general; quasi-isometries need not induce homeomorphisms, even in CAT(0) spaces (Cashen, 2016). A plausible implication is that proportionally contracting rays contribute to a measure-theoretic and probabilistic boundary theory that is less dependent on the topological rigidity of the full contracting boundary.

7. Occurrence, examples, and scope

Proportionally contracting rays arise in groups with contracting elements acting geometrically on proper geodesic spaces. The examples explicitly listed include non-elementary hyperbolic groups, non-elementary relatively hyperbolic groups, CAT(0) cube complexes with rank-one contracting elements, and groups with nontrivial Floyd boundary (Liu et al., 14 Sep 2025). In CAT(0) cube complexes, Morse hyperplanes provide a mechanism for constructing many such rays (Liu et al., 14 Sep 2025).

Among concrete families, right-angled Coxeter groups defined by an irreducible graph with a vertex not in any induced $L$2-cycle furnish examples through their Davis complexes, which admit many Morse hyperplanes and proportionally contracting rays (Liu et al., 14 Sep 2025). The same work states that typical geodesic rays in such spaces are proportionally contracting and that a full-measure subset of the Roller boundary embeds into the Martin boundary (Liu et al., 14 Sep 2025).

A common misconception is to identify proportionally contracting rays with strongly contracting rays. The data do not support such an identification. Strong contraction requires bounded projection diameter along the relevant geodesic or every subsegment, depending on context (Cashen, 2016, Dowdall et al., 2015, Eike et al., 2018). Proportional contraction instead requires that a positive asymptotic proportion of the ray be covered by long uniformly contracting pieces (Liu et al., 14 Sep 2025). Another potential misunderstanding is to interpret the term using unrelated meanings of “proportional” from algebraic geometry, dynamical systems, stochastic geometry, or Euclidean radial maps; those usages concern fiber type contractions, $L$3-contracting systems, restricted blocking rules in Gilbert tessellations, or radial homeomorphisms of the plane, and they are distinct from the geodesic notion under discussion (Fujino et al., 2018, Ofir et al., 2021, Burridge et al., 2012, Georgiou, 12 Jun 2026).

Within geometric group theory, the concept therefore marks a shift from uniform hyperbolicity of individual rays to asymptotic density of hyperbolic-like behavior along typical rays. This suggests a framework in which generic geodesic escape directions can support potential theory and probabilistic boundary identifications even when global Morse behavior is unavailable (Liu et al., 14 Sep 2025).