scholarly journals Upper bounds for the number of resonances on geometrically finite hyperbolic manifolds

2016 ◽  
Vol 18 (5) ◽  
pp. 997-1041 ◽  
Author(s):  
David Borthwick ◽  
Colin Guillarmou
Keyword(s):  
Upper Bounds ◽  
Hyperbolic Manifolds ◽  
Geometrically Finite

scholarly journals Maximally stretched laminations on geometrically finite hyperbolic manifolds

2017 ◽  
Vol 21 (2) ◽  
pp. 693-840 ◽  
Author(s):  
François Guéritaud ◽  
Fanny Kassel
Keyword(s):  
Hyperbolic Manifolds ◽  
Geometrically Finite

scholarly journals Upper bounds for geodesic periods over hyperbolic manifolds

2018 ◽  
Vol 29 (01) ◽  
pp. 1850009
Author(s):  
Feng Su
Keyword(s):  
Upper Bound ◽  
Upper Bounds ◽  
Hyperbolic Manifolds ◽  
Maass Forms ◽  
Maass Form ◽  
Ambient Manifold

We prove an upper bound for geodesic periods of Maass forms over hyperbolic manifolds. By definition, such periods are integrals of Maass forms restricted to a special geodesic cycle of the ambient manifold, against a Maass form on the cycle. Under certain restrictions, the bound will be uniform.

scholarly journals Shrinking targets for the geodesic flow on geometrically finite hyperbolic manifolds

2021 ◽  
Vol 17 (0) ◽  
pp. 401
Author(s):  
Dubi Kelmer ◽  
Hee Oh
Keyword(s):  
Geodesic Flow ◽  
Hyperbolic Manifold ◽  
General Theorem ◽  
Hyperbolic Manifolds ◽  
Quantitative Estimates ◽  
Geometrically Finite ◽  
Logarithm Law ◽  
Tubular Neighborhoods ◽  
Logarithm Laws ◽  
Metric Balls

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \mathscr{M} $\end{document}</tex-math></inline-formula> be a geometrically finite hyperbolic manifold. We present a very general theorem on the shrinking target problem for the geodesic flow, using its exponential mixing. This includes a strengthening of Sullivan's logarithm law for the excursion rate of the geodesic flow. More generally, we prove logarithm laws for the first hitting time for shrinking cusp neighborhoods, shrinking tubular neighborhoods of a closed geodesic, and shrinking metric balls, as well as give quantitative estimates for the time a generic geodesic spends in such shrinking targets.</p>

scholarly journals Upper bounds for geodesic periods over rank one locally symmetric spaces

2018 ◽  
Vol 30 (5) ◽  
pp. 1065-1077 ◽  
Author(s):  
Jan Frahm ◽  
Feng Su
Keyword(s):  
Symmetric Spaces ◽  
Automorphic Forms ◽  
Upper Bounds ◽  
Hyperbolic Manifolds ◽  
Locally Symmetric Spaces ◽  
Ambient Manifold ◽  
Rank One ◽  
General Rank ◽  
Laplace Eigenvalues ◽  
Periods Of Automorphic Forms

AbstractWe prove upper bounds for geodesic periods of automorphic forms over general rank one locally symmetric spaces. Such periods are integrals of automorphic forms restricted to special totally geodesic cycles of the ambient manifold and twisted with automorphic forms on the cycles. The upper bounds are in terms of the Laplace eigenvalues of the two automorphic forms, and they generalize previous results for real hyperbolic manifolds to the context of all rank one locally symmetric spaces.

scholarly journals Excursions of Generic Geodesics in Right-Angled Artin Groups and Graph Products

2019 ◽  
Author(s):  
Yulan Qing ◽  
Giulio Tiozzo
Keyword(s):  
Growth Rate ◽  
Asymptotic Distribution ◽  
Hyperbolic Manifolds ◽  
Artin Groups ◽  
Graph Products ◽  
Counting Measure ◽  
Geometrically Finite ◽  
Right Angled Artin Groups ◽  
Free Product Of Groups ◽  
Product Of Groups

Abstract Motivated by the notion of cusp excursion in geometrically finite hyperbolic manifolds, we define a notion of excursion in any subgroup of a given group and study its asymptotic distribution for right-angled Artin groups (RAAGs) and graph products. In particular, for any irreducible RAAG we show that with respect to the counting measure, the maximal excursion of a generic geodesic in any flat tends to $\log n$, where $n$ is the length of the geodesic. In this regard, irreducible RAAGs behave like a free product of groups. In fact, we show that the asymptotic distribution of excursions detects the growth rate of the RAAG and whether it is reducible.

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