Homological Dimension and Critical Exponent of Kleinian Groups

We prove an inequality between the relative homological dimension of a Kleinian group and its critical exponent. As an application of this result we show that for a geometrically finite Kleinian group Γ, if the topological dimension of the limit set of Γ equals its Hausdorff dimension, then the limit set is a round sphere. Received: March 2007, Revision: October 2007, Accepted: October 2007     

Cores of Hyperbolic 3-Manifolds and Limits of Kleinian Groups II

Troels Jørgensen conjectured that the algebraic and geometriclimits of an algebraically convergent sequence of isomorphicKleinian groups agree if there are no new parabolics in thealgebraic limit. We prove that this conjecture holds in ‘most’cases. In particular, we show that it holds when the domainof discontinuity of the algebraic limit of such a sequence isnon-empty (see Theorem 3.1). We further show, with the sameassumptions, that the limit sets of the groups in the sequenceconverge to the limit set of the algebraic limit. As a corollary,we verify the conjecture for finitely generated Kleinian groupswhich are not (non-trivial) free products of surface groupsand infinite cyclic groups (see Corollary 3.3). These resultsare extensions of similar results for purely loxodromic groupswhich can be found in [4]. Thurston [32] previously establishedthese results in the case when the Kleinian groups are freelyindecomposable (see also Ohshika [24, 25, 27]). Using differenttechniques from ours, Ohshika [26] has proven versions of theseresults for purely loxodromic function groups.     

Convergence Groups, Hausdorff Dimension, and a Theorem of Sullivan and Tukia

We show that a discrete, quasiconformal group preserving ⁿ has the property that its exponent of convergence and the Hausdorff dimension of its limit set detect the existence of a non-empty regular set on the sphere at infinity to ⁿ . This generalizes a result due separately to Sullivan and Tukia, in which it is further assumed that the group act isometrically on ⁿ , i.e. is a Kleinian group. From this generalization we are able to extract geometric information about infinite-index subgroups within certain of these groups.     

Two-Generator Arithmetic Kleinian Groups II

Extending earlier work, we establish the finiteness of the numberof two-generator arithmetic Kleinian groups with one generatorparabolic and the other either parabolic or elliptic. We alsoidentify all the arithmetic Kleinian groups generated by twoparabolic elements. Surprisingly, there are exactly 4 of these,up to conjugacy, and they are all torsion free. 1991 MathematicsSubject Classification 30F40, 20H10.     

Sierpinski锥及其Hausdorff维数与Hausdorff测度

首先给出了 Sierpinski锥的概念及构造过程 ,然后求出其计盒维数、Hausdorff维数和 Hausdorff测度 .     

Embeddings of Kleinian Groups with Torsion

Rips conjectured that a non-clementary word hyperbolic group is cohopfian if and only if it is freely indecomposable. The results and examples in this paper show that cohopficity phenomenon in the case of word hyperbolic group with torsion is much more complicated than the conjecture. In particular, the cohopficity of such groups is not determined by the numbers of their ends and the cohopficity is not preserved by finite index subgroups. Our results and examples arise from Kleinian groups. Orbifold structures and orbifold maps are the new tools in our discussions. Received September, 16, 1999, Accepted September 14, 2000     

Topology of Definable Hausdorff Limits

Let A ^n+r be a set definable in an o-minimal expansion S of the real field, let A ^r be its projection, and assume that the non-empty fibers A_a ⁿ are compact for all a A and uniformly bounded, i.e. all fibers are contained in a ball of fixed radius B(0,R). If L is the Hausdorff limit of a sequence of fibers A_ai, we give an upper-bound for the Betti numbers b_k(L) in terms of definable sets explicitly constructed from a fiber A_a. In particular, this allows us to establish effective complexity bounds in the semialgebraic case and in the Pfaffian case. In the Pfaffian setting, Gabrielov introduced the relative closure to construct the o-minimal structure S_Pfaff generated by Pfaffian functions in a way that is adapted to complexity problems. Our results can be used to estimate the Betti numbers of a relative closure (X,Y)₀ in the special case where Y=.     

拟共形映射和HAUSDORFF维数

讨论了标准Cantor集 F 的Hausdorff维数dim_H(F)在拟共形映射 f 下的变化, 给出了dim_H(f(F))的准确上下界.     

The Hausdorff Dimension of Sections

The notion of finite-type open set condition is defined to calculate the Hausdorff dimensions of the sections of some self-similar sets, such as the dimension of intersection of the Koch curve and the line x = a with a∈Q.     

Hausdorff Dimension and Hausdorff Measures of Julia Sets of Elliptic Functions

It is proved here that if is an elliptic function and q is the maximal multiplicity ofall poles of f, then the Hausdorff dimension of the Julia setof f is greater than 2 q/(q + 1), and the Hausdorff dimensionof the set of points that escape to infinity is less than orequal to 2q/(q + 1). In particular, the area of this latterset is equal to 0. 2000 Mathematics Subject Classification 37F35(primary); 37F10, 30D30 (secondary).     

Hausdorff Dimension and Non-degenerate Families of Projections

We study parameterized families of orthogonal projections for which the dimension of the parameter space is strictly less than that of the Grassmann manifold. We answer the natural question of how much the Hausdorff dimension may decrease by verifying the best possible lower bound for the dimension of almost all projections of a finite measure. We also show that a similar result is valid for smooth families of maps from the n-dimensional Euclidean space to the m-dimensional one.     

RN中某些分形子集的Hausdorff维数与Hausdorff测度

给出了RN中某些分形子集的Hausdorff维数及其Hausdorff测度估计式.     

Hausdorff Dimension of Symmetric Perfect Sets

We consider nowhere dense perfect subsets of [0, 1] that are symmetric but have no additional nice properties. We prove that if E = E_n is a symmetric perfect set and the length of the basic intervals in E_n is denoted by l_n then the Hausdorff dimension of E is

Approximation by Algebraic Integers and Hausdorff Dimension

The paper computes the Hausdorff dimension of sets of real numberswhich are close to infinitely many real algebraic integers ofbounded degree. It also investigates the distribution of realalgebraic integers of bounded degree, which are proved to beevenly spaced.     

Engel连分数展式与Huasdorff维数

本文研究了Engel连分数中部分商以某种速度增长的集合,以及Engel连分数展式收敛速度较快的点组成的集合,利用质量分布原理,证明了这些集合的Haus-dorff维数为1.     

Hausdorff Dimension and Generalized Simultaneous Diophantine Approximation

Suppose that m is a positive integer, is a vector of strictly positive numbers, and Qis an infinite set of positive integers. Let W_Q(m; ) be theset [formula] In this paper we obtain the Hausdorff dimension of this set.We also consider a generalization of the set W_Q(m; ), wherethe error terms in the inequalities are replaced by _i(q), for general functions _i satisfying a certaincondition at infinity. 1991 Mathematics Subject Classification11J83, 28A78.     

Hausdorff Dimension in Convex Bornological Spaces

For non-metrizable spaces the classical Hausdorff dimension is meaningless. We extend the notion of Hausdorff dimension to arbitrary locally convex linear topological spaces and thus to a large class of non-metrizable spaces. This involves a limiting procedure using the canonical bornological structure. In the case of normed spaces the new notion of Hausdorff dimension is equivalent to the classical notion.