Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fractal failures of Ratner rigidity in higher rank geometry

Subhadip Dey, Hee Oh

TL;DR

The article constructs explicit higher-rank fractal counterexamples to Ratner rigidity by introducing floating geodesic planes in SL_3(R)/SO(3) and employing Goldman bulging to produce Zariski-dense Hitchin subgroups with fractal plane closures. It develops a detailed framework based on limit sets in G/P, a nearest-projection map, and careful asymptotic analysis of translated planes to relate fractal dimensions to hyperbolic dynamics, yielding non-integer Hausdorff dimensions that accumulate at 2. The main achievement is the existence of a Γ ≤ SL_3(Z) for which floating planes Y_{L,t} have closures with dimensions strictly between 2 and 3, and can approach 2 from above, highlighting fractal orbit closures in a higher-rank, infinite-volume setting. The work combines precise geometric analysis in higher rank, bulging deformations, and hyperbolic-geometry techniques to illuminate how rigid dichotomies can fail in complex homogeneous spaces.

Abstract

Ratner's theorem shows that in a locally symmetric space of noncompact type and finite volume, every immersed totally geodesic subspace of noncompact type is topologically rigid: its closure is an immersed submanifold. We construct the first explicit higher-rank, infinite-volume examples in which this rigidity fails, via floating geodesic planes. Specifically, we exhibit a Zariski-dense Hitchin surface group $Γ<\mathrm{SL}_3(\mathbb{R})$ such that $Γ\backslash \mathrm{SL}_3(\mathbb{R})/ \mathrm{SO}(3)$ contains a sequence of immersed floating geodesic planes with fractal closures whose Hausdorff dimensions, non-integral, accumulate at $2$. Moreover, $Γ$ can be chosen inside $\mathrm{SL}_3(\mathbb{Z})$. Our method uses Goldman's bulging deformations, but in higher rank new difficulties arise: unlike in rank one, where geodesics orthogonal to a hyperplane always diverge, here one must analyze the collective behavior of entire families of parallel geodesics inside flats under bulging, a phenomenon intrinsic to higher rank.

Fractal failures of Ratner rigidity in higher rank geometry

TL;DR

The article constructs explicit higher-rank fractal counterexamples to Ratner rigidity by introducing floating geodesic planes in SL_3(R)/SO(3) and employing Goldman bulging to produce Zariski-dense Hitchin subgroups with fractal plane closures. It develops a detailed framework based on limit sets in G/P, a nearest-projection map, and careful asymptotic analysis of translated planes to relate fractal dimensions to hyperbolic dynamics, yielding non-integer Hausdorff dimensions that accumulate at 2. The main achievement is the existence of a Γ ≤ SL_3(Z) for which floating planes Y_{L,t} have closures with dimensions strictly between 2 and 3, and can approach 2 from above, highlighting fractal orbit closures in a higher-rank, infinite-volume setting. The work combines precise geometric analysis in higher rank, bulging deformations, and hyperbolic-geometry techniques to illuminate how rigid dichotomies can fail in complex homogeneous spaces.

Abstract

Ratner's theorem shows that in a locally symmetric space of noncompact type and finite volume, every immersed totally geodesic subspace of noncompact type is topologically rigid: its closure is an immersed submanifold. We construct the first explicit higher-rank, infinite-volume examples in which this rigidity fails, via floating geodesic planes. Specifically, we exhibit a Zariski-dense Hitchin surface group such that contains a sequence of immersed floating geodesic planes with fractal closures whose Hausdorff dimensions, non-integral, accumulate at . Moreover, can be chosen inside . Our method uses Goldman's bulging deformations, but in higher rank new difficulties arise: unlike in rank one, where geodesics orthogonal to a hyperplane always diverge, here one must analyze the collective behavior of entire families of parallel geodesics inside flats under bulging, a phenomenon intrinsic to higher rank.

Paper Structure

This paper contains 10 sections, 46 theorems, 243 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Limit sets in G/P
  3. Nearest projection to Y
  4. Floating geodesic planes
  5. Limits of the sequence gamma(t)(s) = a(t) k(0) h(s) o
  6. Nearest projection of floating planes to Y
  7. Closures of floating planes: The Fuchsian case
  8. Bulging deformations and floating planes
  9. Geodesic closures away from a given simple closed geodesic
  10. Orthogonal planes with fractal closures

Key Result

Theorem 1.1

There exists a Zariski dense Hitchin surface subgroup $\Gamma<\operatorname{SL}_3(\mathbb R)$ such that the locally symmetric space $\Gamma\backslash X$ contains a sequence of floating geodesic planes whose closures have Hausdorff dimensions strictly bigger than $2$ and accumulating at $2$. Moreover

Figures (7)

  • Figure 1: Floating geodesic plane
  • Figure 2: Fractal closures of floating geodesic planes
  • Figure 3: The Weyl group.
  • Figure 4: shadow of the floating plane
  • Figure 5: Bulging deformation
  • ...and 2 more figures

Theorems & Definitions (88)

  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2: Limit sets
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 78 more