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The G-Gromov-Hausdorff Distance and Equivariant Topology

Sunhyuk Lim, Facundo Memoli

TL;DR

This work develops a comprehensive framework for G-equivariant stability in metric geometry, introducing the G-Gromov-Hausdorff distance, the G-homotopy type distance, and the G-interleaving distance to compare G-spaces, filtrations, and persistence modules under group actions. It leverages Vietoris-Rips metric thickenings, tight spans, and G-index theory to bound these distances via new G-persistent indices, establishing a hierarchy of invariants that interpolates geometric and topological information in an equivariant setting. Key contributions include a generalized precompactness theorem, rigidity and finiteness results for complete G-Riemannian manifolds, and sharp calculations of d_GH^G between spheres under Z2-actions, including explicit values for Euclidean-sphere variants and square-infinity constructs. The results provide a unified, quantifiable link between geometry and equivariant topology with concrete implications for comparing symmetric spaces and their invariants, enabling sharper lower bounds for G-Gromov-Hausdorff distances and illuminating the role of symmetry in stability phenomena.

Abstract

We made the following changes: (1) reorganized the section order, (2) added a subsection on the Gromov-Hausdorff distance between quotient spaces, and (3) added more examples and remarks.

The G-Gromov-Hausdorff Distance and Equivariant Topology

TL;DR

This work develops a comprehensive framework for G-equivariant stability in metric geometry, introducing the G-Gromov-Hausdorff distance, the G-homotopy type distance, and the G-interleaving distance to compare G-spaces, filtrations, and persistence modules under group actions. It leverages Vietoris-Rips metric thickenings, tight spans, and G-index theory to bound these distances via new G-persistent indices, establishing a hierarchy of invariants that interpolates geometric and topological information in an equivariant setting. Key contributions include a generalized precompactness theorem, rigidity and finiteness results for complete G-Riemannian manifolds, and sharp calculations of d_GH^G between spheres under Z2-actions, including explicit values for Euclidean-sphere variants and square-infinity constructs. The results provide a unified, quantifiable link between geometry and equivariant topology with concrete implications for comparing symmetric spaces and their invariants, enabling sharper lower bounds for G-Gromov-Hausdorff distances and illuminating the role of symmetry in stability phenomena.

Abstract

We made the following changes: (1) reorganized the section order, (2) added a subsection on the Gromov-Hausdorff distance between quotient spaces, and (3) added more examples and remarks.

Paper Structure

This paper contains 41 sections, 61 theorems, 168 equations, 4 figures.

Table of Contents

  1. Introduction.
  2. Acknowledgements
  3. Preliminaries.
  4. Filtrations.
  5. The Vietoris-Rips metric thickening of a metric space.
  6. Tight span of a metric space.
  7. $G$-equivariant distances.
  8. The $G$-Gromov-Hausdorff distance.
  9. $d_\mathrm{GH}^G(X,Y)$ versus $d_\mathrm{GH}(X/G,Y/G)$
  10. The $G$-homotopy type distance.
  11. The proof of \ref{['prop:dHTG']}
  12. The $G$-interleaving distance.
  13. Introducing $G$-group actions.
  14. Hierarchy of $G$-equivariant distances for $\mathrm{VR}^{\mathrm{m}}_p$.
  15. From metric spaces to Vietoris-Rips thickenings.
  16. ...and 26 more sections

Key Result

Theorem 1

There exist homotopy equivalences $\varphi_{r}:\mathrm{VR}_{2r}(X)\rightarrow B_r(X,\mathbf{E}(X))$ for each $r>0$ such that for each $t>s>0$ the following diagram commutes up to homotopy: \begin{tikzcd} \vr_{2r}(X) \arrow[r, hook] \arrow[d, "\varphi_{r}"' , rightarrow] & \vr_{2s}(X) \arrow[d, "\var

Figures (4)

  • Figure 1: Description of $Z$ and $Z/\mathbb{Z}_3$ from \ref{['ex:ghGvsquotientdghHD']}.
  • Figure 2: Description of $(X,d_X)$ from \ref{['ex:dIGdIstrictineq']}.
  • Figure 3: The geodesic $\mathbb{Z}_3$-metric spaces $X$ and $Y$ from \ref{['ex:compatibility']}. The $\mathbb{Z}_3$-action on each of these spaces exchanges different branches leaving the center points fixed.
  • Figure 4: An example of how to build a mapping cylinder neighborhood for the pair $(\Delta_1\times [0,1],(\Delta_1\times{0})\bigcup(\partial\Delta_1\times [0,1]))$. The red region is $A=(\Delta_1\times{0})\bigcup(\partial\Delta_1\times [0,1])$, the green region is the closed neighborhood $N$, and the blue region is $B$.

Theorems & Definitions (190)

  • Definition 1: Filtrations
  • Definition 2: Sub-level set filtrations
  • Definition 3: Vietoris-Rips filtration
  • Definition 4: $p$-diameter
  • Definition 5: $p$-Vietoris-Rips metric thickening filtrations
  • Definition 6: Tight span dress1984treesisbell1964six
  • Theorem 1: lim2020vietoris
  • Definition 7: $G$-set, $G$-topological space, $G$-function, $G$-map, $G$-metric space, and $G$-isometry
  • Example 3.1
  • Definition 8: $G$-correspondence
  • ...and 180 more