Results 1  10
of
52
Group invariant Peano curves
, 1987
"... Our main theorem is that, if M is a closed hyperbolic 3–manifold which fibres over the circle with hyperbolic fibre S and pseudoAnosov monodromy, then the lift of the inclusion of S in M to universal covers extends to a continuous map of B2 to B3, where Bn D Hn n 1 [ S1. The restriction to S 1 1 ma ..."
Abstract

Cited by 41 (2 self)
 Add to MetaCart
Our main theorem is that, if M is a closed hyperbolic 3–manifold which fibres over the circle with hyperbolic fibre S and pseudoAnosov monodromy, then the lift of the inclusion of S in M to universal covers extends to a continuous map of B2 to B3, where Bn D Hn n 1 [ S1. The restriction to S 1 1 maps onto S 2 1 and gives an example of an equivariant S 2 –filling Peano curve. After proving the main theorem, we discuss the case of the figureeight knot complement, which provides evidence for the conjecture that the theorem extends to the case when S is a oncepunctured hyperbolic surface. 20F65; 57M50, 57M60, 57N05, 57N60 1
Automatic structures, rational growth, and geometrically finite hyperbolic groups
 Invent. Math
, 1995
"... Abstract. We show that the set SA(G) of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic group G is dense in the product of the sets SA(P) over all maximal parabolic subgroups P. The set BSA(G) of equivalence classes of biautomatic structures on G is iso ..."
Abstract

Cited by 23 (7 self)
 Add to MetaCart
Abstract. We show that the set SA(G) of equivalence classes of synchronously automatic structures on a geometrically finite hyperbolic group G is dense in the product of the sets SA(P) over all maximal parabolic subgroups P. The set BSA(G) of equivalence classes of biautomatic structures on G is isomorphic to the product of the sets BSA(P) over the cusps (conjugacy classes of maximal parabolic subgroups) of G. Each maximal parabolic P is a virtually abelian group, so SA(P) and BSA(P) were computed in [NS1]. We show that any geometrically finite hyperbolic group has a generating set for which the full language of geodesics for G is regular. Moreover, the growth function of G with respect to this generating set is rational. We also determine which automatic structures on such a group are equivalent to geodesic ones. Not all are, though all biautomatic structures are. 1.
Recognizing constant curvature discrete groups in dimension 3
 TRANS. AMER. MATH. SOC.
, 1998
"... We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3space H 3 in terms of the combinatorics of the action of G on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively cu ..."
Abstract

Cited by 20 (8 self)
 Add to MetaCart
We characterize those discrete groups G which can act properly discontinuously, isometrically, and cocompactly on hyperbolic 3space H 3 in terms of the combinatorics of the action of G on its space at infinity. The major ingredients in the proof are the properties of groups that are negatively curved (in the large) (that is, Gromov hyperbolic), the combinatorial Riemann mapping theorem, and the SullivanTukia theorem on groups which act uniformly quasiconformally on the 2sphere.
A characterization of cocompact hyperbolic and finitevolume hyperbolic groups
 in dimension three, Trans.AMS330 (1992), 419–431. MR 92f:22017
"... Abstract. We show that a cocompact hyperbolic group in dimension 3 is characterized by certain properties of its word metric which depend only on the group structure and not on any action on hyperbolic space. We prove a similar theorem for finitevolume hyperbolic groups in dimension 3. 1. ..."
Abstract

Cited by 19 (2 self)
 Add to MetaCart
Abstract. We show that a cocompact hyperbolic group in dimension 3 is characterized by certain properties of its word metric which depend only on the group structure and not on any action on hyperbolic space. We prove a similar theorem for finitevolume hyperbolic groups in dimension 3. 1.
Random Series In Powers Of Algebraic Integers: Hausdorff Dimension Of The Limit Distribution
, 1995
"... We study the distributions F`;p of the random sums P 1 1 " n` n , where " 1 ; " 2 ; : : : are i.i.d. Bernoullip and ` is the inverse of a Pisot number (an algebraic integer fi whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p = :5, F`;p is a singular m ..."
Abstract

Cited by 18 (0 self)
 Add to MetaCart
We study the distributions F`;p of the random sums P 1 1 " n` n , where " 1 ; " 2 ; : : : are i.i.d. Bernoullip and ` is the inverse of a Pisot number (an algebraic integer fi whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p = :5, F`;p is a singular measure with exact Hausdorff dimension less than 1. We show that in all cases the Hausdorff dimension can be expressed as the top Lyapunov exponent of a sequence of random matrices, and provide an algorithm for the construction of these matrices. We show that for certain fi of small degree, simulation gives the Hausdorff dimension to several decimal places.
Negatively Curved Groups Have The Convergence Property I
 ANNALES ACADEMIAE SCIENTIARUM FENNICAE
, 1995
"... It is known that the Cayley graph \Gamma of a negatively curved (Gromovhyperbolic) group G has a welldefined boundary at infinity @ \Gamma . Furthermore, @ \Gamma is compact and metrizable. In this paper I show that G acts on @ \Gamma as a convergence group. This implies that if @ \Gamma ' S 1 ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
It is known that the Cayley graph \Gamma of a negatively curved (Gromovhyperbolic) group G has a welldefined boundary at infinity @ \Gamma . Furthermore, @ \Gamma is compact and metrizable. In this paper I show that G acts on @ \Gamma as a convergence group. This implies that if @ \Gamma ' S 1 , then G is topologically conjugate to a cocompact Fuchsian group.
Random Walks on Trees with Finitely Many Cone Types
 J. of Theor. Prob
, 2002
"... This paper is devoted to the study of random walks on infinite trees with finitely many cone types (also called periodic trees). We consider nearest neighbour random walks with probabilities adapted to the cone structure of the tree, which include in particular the well studied classes of simple and ..."
Abstract

Cited by 14 (4 self)
 Add to MetaCart
This paper is devoted to the study of random walks on infinite trees with finitely many cone types (also called periodic trees). We consider nearest neighbour random walks with probabilities adapted to the cone structure of the tree, which include in particular the well studied classes of simple and homesick random walks. We give a simple criterion for transience or recurrence of the random walk and prove that the spectral radius is equal to 1 if and only if the random walk is recurrent. Furthermore, we study the asymptotic behaviour of return probabilitites and prove a local limit theorem. In the transient case, we also prove a law of large numbers and compute the rate of escape of the random walk to infinity, as well as prove a central limit theorem. Finally, we describe the structure of the boundary process and explain its connection with the random walk
Local limit theorems for free groups
 MATHEMATISCHE ANNALEN
, 2001
"... In this paper we obtain a local limit theorem for elements of a free group G under the abelianization map [.] : G!/G=[G;G]. This is obtained via an analysis involving subshifts of nite type, where we obtain a result of independent interest. The case of fundamental groups of compact surfaces of genus ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
In this paper we obtain a local limit theorem for elements of a free group G under the abelianization map [.] : G!/G=[G;G]. This is obtained via an analysis involving subshifts of nite type, where we obtain a result of independent interest. The case of fundamental groups of compact surfaces of genus >= 2 is also discussed.
Formal Language Theory And The Geometry Of 3Manifolds
 Commentarii Math. Helv
, 1996
"... . Automatic groups were introduced in connection with geometric problems, in particular with the study of fundamental groups of 3manifolds. In this article the class of automatic groups is extended to include the fundamental group of every compact 3manifold which satisfies Thurston's geometrizatio ..."
Abstract

Cited by 12 (8 self)
 Add to MetaCart
. Automatic groups were introduced in connection with geometric problems, in particular with the study of fundamental groups of 3manifolds. In this article the class of automatic groups is extended to include the fundamental group of every compact 3manifold which satisfies Thurston's geometrization conjecture. Toward this end the class C A of asynchronously Acombable groups is introduced and studied, where A is an arbitrary full abstract family of languages. For example A may be the family of regular languages Reg, contextfree languages CF, or indexed languages Ind.TheclassC Reg consists of precisely those groups which are asynchronously automatic. It is proved that C Ind contains all of the above fundamental groups, but that C CF does not. Indeed a virtually nilpotent group belongs to C CF if and only if it is virtually abelian. Introduction In the last several years there has been a remarkable interplay between topology, geometry, group theory, and the t...
Comparison Theorems And Orbit Counting In Hyperbolic Geometry
 Trans. Amer. Math. Soc
, 1998
"... In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a mixture of ..."
Abstract

Cited by 11 (11 self)
 Add to MetaCart
In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a mixture of ideas from both "thermodynamic" ergodic theory and the automaton associated to strongly Markov groups. 0.