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].

Geometrically finite groups, Khintchine-type theorems and Hausdorff dimension

1996 ◽  
Vol 120 (4) ◽  
pp. 647-662 ◽  
Author(s):  
Sanju L. Velani
Keyword(s):  
Hyperbolic Space ◽  
Finite Groups ◽  
Discrete Subgroup ◽  
Dimensional Euclidean Space ◽  
Simultaneous Diophantine Approximation ◽  
Geometrically Finite ◽  
Geometrically Finite Group ◽  
Fundamental Polyhedron ◽  
Image Position ◽  
Geometrically Finite Groups

1·1. Groups of the first kind. In [11], Patterson proved a hyperbolic space analogue of Khintchine's theorem on simultaneous Diophantine approximation. In order to state Patterson's theorem, some notation and terminology are needed. Let ‖x‖ denote the usual Euclidean norm of a vector x in k+1, k + 1-dimensional Euclidean space, and let be the unit ball model of k + 1-dimensional hyperbolic space with Poincaré metric ρ. A non-elementary geometrically finite group G acting on Bk + 1 is a discrete subgroup of Möb (Bk+l), the group of orientation preserving Mobius transformations preserving Bk + 1, for which there exists some convex fundamental polyhedron with finitely many faces. Since G is non-elementary, the limit set L(G) of G – the set of limit points in the unit sphere Sk of any orbit of G in Bk+1 – is uncountable. The group G is said to be of the first kind if L(G) = Sk and of the second kind otherwise.

scholarly journals NONVANISHING OF ALGEBRAIC ENTROPY FOR GEOMETRICALLY FINITE GROUPS OF ISOMETRIES OF HADAMARD MANIFOLDS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 799-813
Author(s):  
ROGER C. ALPERIN ◽  
GENNADY A. NOSKOV
Keyword(s):  
Finite Groups ◽  
Hadamard Manifold ◽  
Hadamard Manifolds ◽  
Algebraic Entropy ◽  
Geometrically Finite ◽  
Geometrically Finite Group ◽  
Group Of Isometries ◽  
Groups Of Isometries ◽  
Uniform Exponential Growth ◽  
Geometrically Finite Groups

We prove that any nonelementary geometrically finite group of isometries of a pinched Hadamard manifold has nonzero algebraic entropy in the sense of M. Gromov. In other words it has uniform exponential growth.

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 ON A GENERALIZATION OF DEHN'S ALGORITHM

2008 ◽  
Vol 18 (07) ◽  
pp. 1137-1177 ◽  
Author(s):  
OLIVER GOODMAN ◽  
MICHAEL SHAPIRO
Keyword(s):  
Finite Groups ◽  
Nilpotent Groups ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Closure Properties ◽  
Relatively Hyperbolic Groups ◽  
Geometrically Finite ◽  
Geometrically Finite Groups ◽  
Relatively Hyperbolic ◽  
Infinite Subgroup

Viewing Dehn's algorithm as a rewriting system, we generalize to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include finitely generated nilpotent groups, many relatively hyperbolic groups including geometrically finite groups and fundamental groups of certain geometrically decomposable 3-manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm. We dub these Cannon's algorithms.

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