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 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 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.

scholarly journals On interpreting Patterson–Sullivan measures of geometrically finite groups as Hausdorff and packing measures

2015 ◽  
Vol 36 (8) ◽  
pp. 2675-2686
Author(s):  
DAVID SIMMONS
Keyword(s):  
New York ◽  
Finite Groups ◽  
Acta Math ◽  
Hyperbolic Manifolds ◽  
Packing Measure ◽  
Geometrically Finite ◽  
Logarithm Law ◽  
Conformal Measures ◽  
Finite Kleinian Group ◽  
Geometrically Finite Groups

We provide a new proof of a theorem whose proof was sketched by Sullivan [Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math.149(3–4) (1982), 215–237], namely that if the Poincaré exponent of a geometrically finite Kleinian group $G$ is strictly between its minimal and maximal cusp ranks, then the Patterson–Sullivan measure of $G$ is not proportional to the Hausdorff or packing measure of any gauge function. This disproves a conjecture of Stratmann [Multiple fractal aspects of conformal measures; a survey. Workshop on Fractals and Dynamics (Mathematica Gottingensis, 5). Eds. M. Denker, S.-M. Heinemann and B. Stratmann. Springer, Berlin, 1997, pp. 65–71; Fractal geometry on hyperbolic manifolds. Non-Euclidean Geometries (Mathematical Applications (N.Y.), 581). Springer, New York, 2006, pp. 227–247].

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