Quasi-Redirecting Boundaries in Metric Geometry

• Quasi-Redirecting Boundaries are geometric and analytic phenomena that demarcate the interface between nonpositive curvature and generalized metric structures.
• They are central to understanding coarse embedding obstructions, flat plane rigidity via convex bicombings, and subtle volume growth anomalies in nonpositively curved spaces.
• These boundaries inform the stability of geometric flows and provide precise analytic and geometric thresholds crucial for rigidity in both finite and infinite-dimensional settings.

Quasi-Redirecting Boundaries are geometric, analytic, and large-scale phenomena that arise in the study of nonpositive curvature, coarse geometry, and the interplay between local and global metric structure. While the term is not standard, the component concepts are central to several deep results and methods in modern metric geometry—including the existence and rigidity of flats in spaces admitting generalized convex bicombings, the behavior of asymptotic volume growth under nonpositive curvature, sharp coarse embedding obstructions, and Lagrangian stability in infinite-dimensional groups endowed with right-invariant metrics. These boundaries or "almost-flat" or "almost-hyperbolic-to-flat" structures play a pivotal role in distinguishing nonpositive from nonnegative curvature at scale, in the rigidity theory of generalized spaces, and in the stability properties of geometric flows.

1. Coarse Embeddings, Nonpositive Curvature, and Metric Cotype

Let $(Y, d_Y)$ and $(X, d_X)$ be metric spaces. A map $f: Y \to X$ is a coarse embedding if there exist nondecreasing functions $\omega, \Omega$ with $\omega(t) \to \infty$ as $t\to\infty$ such that

$$\omega(d_Y(y, y')) \le d_X(f(y), f(y')) \le \Omega(d_Y(y, y')),\quad \forall y,y'\in Y.$$

This encodes large-scale geometric similarity, disregarding finer scales. Alexandrov spaces of nonpositive curvature (CAT(0) spaces) provide a canonical geometric model; a CAT(0) space $(X,d_X)$ satisfies, for all midpoints $w$ of $x,y$ and any $z\in X$,

$$d_X(z,w)^2 + \tfrac{1}{4} d_X(x,y)^2 \le \tfrac{1}{2} d_X(z,x)^2 + \tfrac{1}{2} d_X(z,y)^2.$$

The existence, or obstruction, of coarse embeddings of a space $Y$ into a CAT(0) space reveals fundamental geometric boundaries. Eskenazis, Mendel, and Naor proved that $\ell_p$ for any $p>2$ does not coarse-embed into any CAT(0) space, in contrast to the “coarse universality” of Alexandrov spaces of nonnegative curvature—every metric space coarsely embeds into some such space (Eskenazis et al., 2018).

A key analytic boundary is given by sharp metric cotype: a $q$–barycentric space $(X,d_X)$ with constant $\beta$ satisfies a nonlinear martingale-type inequality that combines barycenter maps and metric cotype, which serves as a large-scale invariant obstructing coarse embeddings from spaces with higher cotype.

2. Flat Planes and Bicombings: Geodesic Convexity Generalizations

A central feature of nonpositively curved spaces is the existence and structure of flat subspaces. In spaces possibly lacking unique geodesics, “convex geodesic bicombings” select for any $x,y\in X$ a constant-speed geodesic $\sigma_{xy}$ satisfying reversibility and a form of conical convexity,

$$d(\sigma_{xy}(t), \sigma_{x'y'}(t)) \le (1-t) d(x,x') + t d(y,y').$$

Consistency is an additional property ensuring that subsegments of $\sigma_{xy}$ are themselves images under the bicombing.

The generalized flat-plane theorem asserts that, under cocompact group actions and existence of a consistent bicombing, the presence of an isometrically embedded normed $\mathbb{R}^2$ (a norm-flat) is equivalent to non-hyperbolicity; in other words, the failure to have a quasi-redirecting boundary (i.e., the absence of such flats) enforces Gromov–hyperbolicity (Descombes et al., 2015).

This principle underlies the Flat Torus Theorem: if $X$ has a consistent bicombing and $\Gamma$ acts properly and cocompactly, any free abelian subgroup $A\cong \mathbb{Z}^n$ corresponds to an isometrically embedded $\mathbb{R}^n$-norm flat, providing a canonical quasi-flat boundary.

3. Rigidity Through Volume Asymptotics and Margulis Function

In rank-one nonpositively curved manifolds $(M,g)$, the Riemannian volume of large balls in the universal cover $X$ grows as

$$\lim_{t \to \infty} \frac{b_t(x)}{e^{ht}/h} = c(x),$$

where $h$ is the topological entropy of the geodesic flow, and $c(x)$ is the Margulis function (Wu, 2021). The $C^1$ regularity of $c(x)$ ensures a strong form of regularity for quasi-redirecting boundaries in the volume growth sense.

For closed rank-one surfaces without flat strips, $c(x)$ is constant if and only if the curvature is constant negative, implying that deviation from constant $c(x)$ marks the presence of quasi-flat or non-hyperbolic boundaries. More generally, flip-invariance of Patterson–Sullivan measures and conditional measures on fiber bundles is rigidly tied to local symmetry, with its failure indicating the persistence of quasi-redirecting or non-symmetric boundaries.

4. Obstruction Theorems and Bi-Lipschitz Distortion

Sharp metric cotype inequalities provide analytic “boundaries”—spaces such as $\ell_p$ do not embed coarsely into any $q$–barycentric (including CAT(0)) spaces if $p > q$ (Eskenazis et al., 2018). This analytic boundary is quantified through bi-Lipschitz distortion of discrete grids:

• The distortion for $[m]^n_\infty$ into $\ell_q$ is $\asymp \min\{n^{1/q}, m\}$ for $q>2$.
• Classical results (Ribe rigidity) further show that $q \gtrsim \log n$ is needed for low-distortion embeddings of combinatorial cubes, thus creating a precise coarse geometric boundary in metric embedding theory.

These results underscore the role of sharp invariants—houses of quasi-redirecting boundaries—determinant for embeddings and rigidity across classes of spaces.

5. Convexity, Pathologies, and Generalized Nonpositive Curvature Phenomena

In geometric settings lacking unique geodesics or the full CAT(0) inequality, the structure of bicombings (conical, reversible, consistent) determines whether quasi-redirecting boundaries persist. Examples demonstrate that weakening convexity can result in non-unique thickness for strips between lines, absence of $\sigma$–axes, or non-convex images of embedded normed planes (Descombes et al., 2015). These pathologies outline the limits of flat-embedding results and demarcate the boundary between generalized nonpositive curvature and hyperbolicity.

CAT(0) and Busemann spaces possess unique (or all) geodesic convexity, respectively, providing a full suite of geometric tools. In more general bicombing frameworks, unique geodesics are replaced by canonicity and selectivity, but many large-scale phenomena—flats, flat-torus theorem, and hyperbolicity criteria—still rely on these convexity boundaries.

6. Infinite-Dimensional Nonpositive Curvature and Lagrangian Stability

In analysis on infinite-dimensional groups, such as the quantomorphism group $\mathcal{D}_q(M)$, nonpositive curvature manifests in the study of geodesic flows governed by equations of physical significance, e.g., the quasi-geostrophic equation in geophysics (Lee et al., 2019). Arnold's curvature formula, with specialization to steady shear flows, yields explicit nonpositivity criteria in terms of physical quantities (Froude and Rossby numbers).

A sharp boundary is defined by critical values $Fr_c(Ro)$ and $Ro_c(Fr)$ for which the sectional curvature remains nonpositive. Invariant shearing directions and regularization by the $H^1$-type metric ensure that, within the nonpositive curvature regime, Lagrangian stability is enhanced. Limiting cases, such as the Euler and $f$-plane scenarios, illustrate the persistence or weakening of these boundaries across physical models.

7. Open Directions and Large-Scale Geometric Boundaries

The contrast between nonnegative and nonpositive curvature in terms of coarse universality encapsulates a vast geometric boundary: every metric space embeds coarsely into some Alexandrov space of nonnegative curvature, but key analytic and geometric invariants strictly obstruct such embeddings into spaces of nonpositive curvature (Eskenazis et al., 2018).

Open questions remain regarding the identification of bounded-geometry spaces not embeddable into any CAT(0) space, the universal properties of CAT(0) manifolds themselves, and the delineation between stable metric spaces and the full spectrum of large-scale geometric boundaries.

The study of quasi-redirecting boundaries thus inhabits the intersection of geometric group theory, infinite-dimensional analysis, convex geometry, and dynamical rigidity, capturing precise thresholds—analytic and geometric—at which global nonpositive curvature imparts rigidity, obstruction, or flexibility to the underlying space.