RECORD
On Borel Anosov subgroups of ${\rm SL}(d,\mathbb{R})$
Geometric Topology · Submitted 3 Aug 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
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 Andr\'es Sambarino. Furthermore, we show restrictions on the hyperbolic spaces admitting uniformly regular quasi-isometric embeddings into the symmetric space $X_d$ of ${\rm SL}(d,\mathbb{R})$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We study the antipodal subsets of the full flag manifolds $\mathcal{F}(\mathbb{R}^d)$. | contribution 0.95 | 0.73 | 0.03 | asserted 0.51 |
| s1 | 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. | result 1.00 | 0.96 | 0.84 | asserted 1.00 |
| s2 | This gives a partial answer to a question asked by Andr\'es Sambarino. | interpretation 0.82 | 0.43 | 0.04 | asserted 0.92 |
| s3 | Furthermore, we show restrictions on the hyperbolic spaces admitting uniformly regular quasi-isometric embeddings into the symmetric space $X_d$ of ${\rm SL}(d,\mathbb{R})$. | result 0.95 | 0.94 | 0.08 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v3
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original 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 Andr\'es Sambarino. Furthermore, we show restrictions on the hyperbolic spaces admitting uniformly regular quasi-isometric embeddings into the symmetric space $X_d$ of ${\rm SL}(d,\mathbb{R})$.
Human review
How the literature uses it
3 arXiv papers cite this in 12 sentences
Papers citing it · 8 sentences without a confident label
Builds on it 1 paper
-
The proof of <Ref>(i) is based on extending the techniques of [ref] to partial flag manifolds in general, which we do in [ref], supplemented with an elementary observation about partial flag manifolds associated to $(2n,)$; see [ref].
Restrictions on Anosov subgroups of Sp(2n,R) · Outline of the paper · 0.99
Uses its method or tool 1 paper
-
Since $_Θ$ has Property (I), by the same argument used in the proof of [ref] (see [ref] ), it follows that every point in $Ω$ is not antipodal to some point in $f(-1,1)$.
Restrictions on Anosov subgroups of Sp(2n,R) · Restrictions on Anosov subgroups · 0.85
Compares with it 1 paper
-
For $𝕂=ℝ$ and $d=2p$, the previous previous theorem recovers the main result of Dey [ref] in dimension $2p$ and of the second named author [ref] .
On projective Anosov subgroups of symplectic groups · Introduction · 0.89
Cites it as background 1 paper
-
The question of finding topological restrictions on groups that admit Anosov representations in higher rank Lie groups has been investigated, e.g., by Canary–Tsouvalas [ref] , Dey [ref] , Tsouvalas [ref] for $SL(d,R)$, Dey–Greenberg–Riestenberg [ref] for $Sp(2d,R)$, and Pozzetti–Tsouvalas [ref] for $Sp(2d,C)$.
Topological and geometric restrictions on hyperconvex representations · Introduction · 0.97
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.