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

Let $G=\operatorname{SO}^\circ(n,1) \times \operatorname{SO}^\circ(n,1)$ and$X={\mathbb H}^{n}\times {\mathbb H}^{n}$ for $n\ge 2$. For a pair $(\pi_1,\pi_2)$ of non-elementary convex cocompact representations of a finitelygenerated group $\Sigma$ into $\operatorname{SO}^\circ(n,1)$, let$\Gamma=(\pi_1\times \pi_2)(\Sigma)$. Denoting the bottom of the $L^2$-spectrumof the negative Laplacian on $\Gamma\backslash X$ by $\lambda_0$, we show:  (1) $L^2(\Gamma\backslash G)$ is tempered and $\lambda_0=\frac{1}{2}(n-1)^2$;  (2) There exists no positive Laplace eigenfunction in $L^2(\Gamma \backslashX)$.  In fact, analogues of (1)-(2) hold for any Anosov subgroup $\Gamma$ in theproduct of at least two simple algebraic groups of rank one as well as forHitchin subgroups $\Gamma<\operatorname{PSL}_d(\mathbb R)$, $d\ge 3$. Moreover,if $G$ is a semisimple real algebraic group of rank at least $2$, then (2)holds for any Anosov subgroup $\Gamma$ of $G$.

Temperedness of $L^2(\Gamma\backslash G)$ and positive eigenfunctions in higher rank