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On Borel Anosov subgroups of ${\rm SL}(d,\mathbb{R})$

Subhadip Dey

TL;DR

The paper addresses the problem of understanding which Borel Anosov subgroups of $\mathrm{SL}(d,\mathbb{R})$ can occur, by analyzing antipodal subsets of the full flag manifold $\mathcal{F}(\mathbb{R}^d)$. It introduces a central involution on the intersection of opposite maximal Schubert cells and proves a swaps theorem that prohibits invariant components in most dimensions, enabling a global obstruction result. Consequently, for $d \ge 2$ with $d \neq 5$ and $d \equiv 2,3,4,5,$ or $6 \pmod{8}$, Borel Anosov subgroups are shown to be virtually free or surface groups, partially answering Sambarino’s question, and it derives restrictions on uniformly regular quasi-isometric embeddings into the symmetric space $X_d$. The work unifies dynamics on flag varieties with hyperbolic group theory to constrain the possible limit sets and their geometric embedding properties, with particular attention to the case $d=5$ and potential extensions. Overall, it advances the understanding of the interface between higher rank Anosov dynamics, flag geometry, and coarse embeddings, providing both structural theorems and new obstructions.

Abstract

We study the antipodal subsets of the full flag manifolds $\mathcal{F}(\mathbb{R}^d)$. As a consequence, for natural numbers $d \ge 2$ such that $d\ne 5$ and $d \not\equiv 0,\pm1 \mod 8$, we show that Borel Anosov subgroups of ${\rm SL}(d,\mathbb{R})$ are virtually isomorphic to either a free group or the fundamental group of a closed hyperbolic surface. This gives a partial answer to a question asked by Andrés Sambarino. Furthermore, we show restrictions on the hyperbolic spaces admitting uniformly regular quasi-isometric embeddings into the symmetric space $X_d$ of ${\rm SL}(d,\mathbb{R})$.

On Borel Anosov subgroups of ${\rm SL}(d,\mathbb{R})$

TL;DR

The paper addresses the problem of understanding which Borel Anosov subgroups of can occur, by analyzing antipodal subsets of the full flag manifold . It introduces a central involution on the intersection of opposite maximal Schubert cells and proves a swaps theorem that prohibits invariant components in most dimensions, enabling a global obstruction result. Consequently, for with and or , Borel Anosov subgroups are shown to be virtually free or surface groups, partially answering Sambarino’s question, and it derives restrictions on uniformly regular quasi-isometric embeddings into the symmetric space . The work unifies dynamics on flag varieties with hyperbolic group theory to constrain the possible limit sets and their geometric embedding properties, with particular attention to the case and potential extensions. Overall, it advances the understanding of the interface between higher rank Anosov dynamics, flag geometry, and coarse embeddings, providing both structural theorems and new obstructions.

Abstract

We study the antipodal subsets of the full flag manifolds . As a consequence, for natural numbers such that and , we show that Borel Anosov subgroups of are virtually isomorphic to either a free group or the fundamental group of a closed hyperbolic surface. This gives a partial answer to a question asked by Andrés Sambarino. Furthermore, we show restrictions on the hyperbolic spaces admitting uniformly regular quasi-isometric embeddings into the symmetric space of .
Paper Structure (14 sections, 19 theorems, 62 equations, 1 figure, 1 table)

This paper contains 14 sections, 19 theorems, 62 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Antipodal subsets
  3. Borel Anosov subgroups
  4. Uniformly regular quasi-isometric embeddings
  5. An involution on the intersection of two opposite maximal Schubert cells
  6. Proof of Theorem \ref{['thm:swaps']} when $d=3$
  7. Proof of Theorem \ref{['thm:swaps']} when $d \equiv 2 \mod 4$
  8. Some preparation before the proof of Theorem \ref{['thm:swaps']} in the remaining cases
  9. Proof of Theorem \ref{['thm:swaps']} when $d=4$
  10. Proof of Theorem \ref{['thm:swaps']} when $d$ is odd and $d\ge 7$
  11. Proof of Theorem \ref{['thm:swaps']} when $d \equiv 4\mod 8$ and $d\ge 12$
  12. Proof of Theorem \ref{['thm:main']}
  13. Proofs of Theorems \ref{['thm:main0']} and \ref{['thm:main1']}
  14. Some further remarks

Key Result

Theorem 1

Let $d$ be any natural number satisfying eqn:d. Let $\sigma_\pm\in \mathcal{F}_d$ be any pair of antipodal points, and let $\Omega$ be any connected component of $\mathcal{F}_d \setminus (\mathcal{E}_{\sigma_-} \cup \mathcal{E}_{\sigma_+}) = \mathcal{C}_{\sigma_-} \cap \mathcal{C}_{\sigma_+}$. If $c then, for all $\sigma\in \Omega$, the image of $c$ intersects $\mathcal{E}_\sigma$.

Figures (1)

  • Figure 1: The part of the set $\mathcal{E}_{\sigma_+}$ lying in $\mathbf{R}^3 \cong \mathcal{C}_{\sigma_-} \subset \mathcal{F}_3$. The six components of $\mathcal{C}_{\sigma_-}\cap \mathcal{C}_{\sigma_+}$ are visible in the complement of this algebraic surface.

Theorems & Definitions (44)

  • Theorem 1
  • Remark 1.1
  • Definition 1.2: Antipodal subsets and maps
  • Corollary 2
  • proof
  • Definition 1.3: Boundary embedded subgroups
  • Theorem 3
  • Definition 1.4: Borel Anosov subgroups
  • Corollary 4
  • Remark 1.5
  • ...and 34 more