English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In the paper "Pappus's theorem and the modular group", R. Schwartzconstructed a 2-dimensional family of faithful representations $\rho_\Theta$ ofthe modular group $\mathrm{PSL}(2,\mathbb{Z})$ into the group $\mathscr{G}$ ofprojective symmetries of the projective plane via Pappus Theorem. The image ofthe unique index 2 subgroup $\mathrm{PSL}(2,\mathbb{Z})_o$ of$\mathrm{PSL}(2,\mathbb{Z})$ under each representation $\rho_\Theta$ is in thesubgroup $\mathrm{PGL}(3,\mathbb{R})$ of $\mathscr{G}$ and preserves atopological circle in the flag variety, but $\rho_\Theta$ is not Anosov. In herPhD Thesis, V. P. Val\'erio elucidated the Anosov-like feature of Schwartzrepresentations: For every $\rho_\Theta$, there exists a 1-dimensional familyof Anosov representations $\rho^\varepsilon_{\Theta}$ of$\mathrm{PSL}(2,\mathbb{Z})_o$ into $\mathrm{PGL}(3,\mathbb{R})$ whose limit isthe restriction of $\rho_\Theta$ to $\mathrm{PSL}(2,\mathbb{Z})_o$. In thispaper, we improve her work: For each $\rho_\Theta$, we build a 2-dimensionalfamily of Anosov representations of $\mathrm{PSL}(2,\mathbb{Z})_o$ into$\mathrm{PGL}(3,\mathbb{R})$ containing $\rho^\varepsilon_{\Theta}$ and a1-dimensional subfamily of which can extend to representations of$\mathrm{PSL}(2,\mathbb{Z})$ into $\mathscr{G}$. Schwartz representations aretherefore, in a sense, the limits of Anosov representations of$\mathrm{PSL}(2,\mathbb{Z})$ into $\mathscr{G}$.

Pappus Theorem, Schwartz Representations and Anosov Representations