Equivalence of Gromov Boundary and Metric Boundary in Quasihyperbolically Visible Spaces

Core Concepts

Under certain conditions, a space's Gromov boundary and metric boundary are equivalent if and only if the space is quasihyperbolically visible, meaning quasihyperbolic geodesics connecting sequences converging to distinct boundary points must intersect a common compact set.

Abstract

Allu, V. & Pandey, A. (2024). A note on the equivalence of Gromov boundary and metric boundary. arXiv. arXiv:2405.10273v2 [math.MG]

This paper investigates the conditions under which the Gromov boundary and the metric boundary of a minimally nice metric space are equivalent. The authors aim to address the open problem of determining when the identity map from a space equipped with the quasihyperbolic metric to the same space with its original metric extends continuously or homeomorphically to the respective boundaries.

Key Insights Distilled From

A note on the equivalence of Gromov boundary and metric boundary

by Vasudevarao ... at arxiv.org 11-19-2024

https://arxiv.org/pdf/2405.10273.pdf

Deeper Inquiries

Can the results regarding the equivalence of Gromov and metric boundaries be extended to more general classes of metric spaces beyond bounded minimally nice spaces?

Extending the results about the equivalence of Gromov and metric boundaries to more general metric spaces beyond bounded minimally nice spaces is a challenging but potentially fruitful endeavor. Here's a breakdown of the challenges and possible approaches:
Challenges:

Lack of Geodesics: Minimally nice spaces, by definition, are rectifiably connected and hence admit a geodesic metric. General metric spaces may not have this property, making it difficult to define the Gromov boundary, which relies on the notion of geodesic rays.
Absence of Quasihyperbolic Metric: The quasihyperbolic metric, a crucial tool in this context, is defined using the distance to the metric boundary. In spaces where the metric boundary is not well-behaved or doesn't capture the "infinity" of the space adequately, defining a meaningful quasihyperbolic metric becomes problematic.
Weaker Control over Geometry: Boundedness and the minimally nice property impose a certain degree of control over the geometry of the space. Relaxing these conditions might lead to situations where the interplay between the Gromov boundary (capturing large-scale geometry) and the metric boundary (reflecting the completion) becomes more intricate.
Possible Approaches and Considerations:

Generalizations of Hyperbolicity: Explore notions of hyperbolicity that don't rely on geodesics, such as notions based on Gromov's four-point condition or other equivalent definitions. This could allow for the definition of a "boundary at infinity" even in the absence of a traditional Gromov boundary.
Alternative Intrinsic Metrics: Instead of the quasihyperbolic metric, investigate other intrinsic metrics that might be more suitable for the specific class of metric spaces under consideration. These metrics should ideally capture the large-scale geometry of the space in a way that relates meaningfully to a suitable notion of a boundary.
Restricting to Subclasses: Focus on specific classes of metric spaces that possess additional structure or properties that could facilitate the extension of the results. For example, one might consider spaces with a well-defined boundary at infinity, even if they are not Gromov hyperbolic in the classical sense.
In summary: Extending the equivalence results requires navigating the absence of key structures present in minimally nice spaces. Success likely hinges on carefully selecting appropriate generalizations of hyperbolicity and boundary notions, potentially alongside restrictions to subclasses of metric spaces where these generalizations are well-behaved.

Could there be alternative geometric conditions, other than quasihyperbolic visibility, that also guarantee the equivalence of these boundaries?

Yes, it's plausible that alternative geometric conditions, besides quasihyperbolic visibility, could guarantee the equivalence of Gromov and metric boundaries. Here are some potential avenues for exploration:
1. Conditions Related to Geodesic Convexity:

Stronger Convexity Properties: Instead of quasi-convexity, explore the implications of stronger convexity properties, such as strict convexity or uniform convexity (in the context of geodesic metric spaces). These stronger conditions might provide tighter control over the behavior of geodesics and their convergence to the boundary.
Gromov Product Conditions: Investigate conditions involving the Gromov product, a key tool in Gromov hyperbolic spaces. For instance, one could examine conditions that relate the Gromov product of boundary points to the behavior of sequences converging to those points in the metric space.
2. Conditions Involving Boundary Structure:

Boundary Accessibility: Explore conditions that guarantee a certain degree of accessibility of the metric boundary from within the space. For example, one might require that every boundary point is the limit of a quasi-geodesic ray.
Boundary Regularity: Investigate the implications of imposing regularity conditions on the metric boundary, such as requiring it to be a topological sphere or a space with controlled topological dimension. Such conditions might impose constraints on the behavior of sequences converging to the boundary.
3. Analytic Conditions:

Growth Conditions on Other Metrics: Instead of focusing solely on the quasihyperbolic metric, explore growth conditions on other relevant metrics, such as visual metrics or metrics defined using suitable potential functions.
Poincaré Inequalities: In the context of spaces equipped with a measure, investigate the implications of Poincaré-type inequalities, which relate the integral of a function to the integral of its gradient. These inequalities can provide control over the oscillation of functions and might be used to study the behavior of sequences converging to the boundary.
In essence: The key is to identify geometric or analytic conditions that provide sufficient control over the asymptotic behavior of the space, ensuring that sequences converging to the same point in the Gromov boundary also converge to the same point in the metric boundary, and vice versa.

How can the insights gained from understanding the relationship between Gromov and metric boundaries be applied to solve problems in related fields like geometric group theory or the study of hyperbolic groups?

The relationship between Gromov and metric boundaries has significant implications for geometric group theory and the study of hyperbolic groups. Here's how these insights can be applied:
1. Understanding Group Boundaries:

Visualizing Group Structure: For hyperbolic groups, the Gromov boundary provides a powerful way to visualize the "infinity" of the group. Equivalence results between the Gromov boundary and other naturally occurring boundaries (like the boundary of a Cayley graph) give concrete geometric interpretations of the group's structure at infinity.
Classifying Groups: The topology and geometry of the Gromov boundary are quasi-isometry invariants, meaning they are preserved under distortions that don't change the large-scale geometry. This makes the boundary a valuable tool for classifying and distinguishing between different hyperbolic groups.
2. Analyzing Group Actions:

Convergence Actions: Hyperbolic groups act naturally on their Gromov boundaries by homeomorphisms. Understanding the dynamics of these actions provides insights into the algebraic properties of the group. The equivalence of boundaries can help translate these dynamics to other settings where the metric boundary might be more natural.
Rigidity Results: In some cases, the action of a group on its boundary is rigid, meaning that any continuous map of the boundary that "almost" commutes with the group action is close to an actual group element. Such rigidity results have profound implications for understanding the structure of the group and its subgroups.
3. Applications to Geometric Problems:

Solving Isomorphism Problems: The structure of the Gromov boundary can be used to solve isomorphism problems for certain classes of groups. For example, two hyperbolic groups with non-homeomorphic boundaries cannot be isomorphic.
Studying Quasi-Isometries: The Gromov boundary is a quasi-isometry invariant, making it a useful tool for studying quasi-isometries between groups and spaces. For instance, one can use the boundary to show that two spaces are not quasi-isometric by demonstrating that their boundaries are not homeomorphic.
In summary: The interplay between Gromov and metric boundaries provides a bridge between the large-scale geometry of groups and spaces and their concrete geometric realizations. This connection has led to significant advances in geometric group theory, particularly in the study of hyperbolic groups, and continues to be an active area of research.