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

We consider the $SL(2,R)$ action on moduli spaces of quadratic differentials.If $\mu$ is an $SL(2,R)$-invariant probability measure, crucial informationabout the associated representation on $L^2(\mu)$ (and in particular, fineasymptotics for decay of correlations of the diagonal action, the Teichm\"ullerflow) is encoded in the part of the spectrum of the corresponding foliatedhyperbolic Laplacian that lies in $(0,1/4)$ (which controls the contribution ofthe complementary series). Here we prove that the essential spectrum of aninvariant algebraic measure is contained in $[1/4,\infty)$, i.e., for every$\delta>0$, there are only finitely many eigenvalues (counted withmultiplicity) in $(0,1/4-\delta)$. In particular, all algebraic invariantmeasures have a spectral gap.

Small eigenvalues of the Laplacian for algebraic measures in moduli space, and mixing properties of the Teichm\"uller flow