Results 1  10
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50
A rigidity theorem for the solvable BaumslagSolitar groups
, 1996
"... this paper we take the first steps towards applying some of these ideas to proving rigidity results for groups that arise most naturally not in geometry but in combinatorial group theory. ..."
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Cited by 78 (11 self)
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this paper we take the first steps towards applying some of these ideas to proving rigidity results for groups that arise most naturally not in geometry but in combinatorial group theory.
On the spectrum of Hecke type operators related to some fractal groups
 TRUDY MAT. INST. STEKLOV
, 1999
"... We give the first example of a connected 4regular graph whose Laplace operator’s spectrum is a Cantor set, as well as several other computations of spectra following a common “finite approximation” method. These spectra are simple transforms of the Julia sets associated to some quadratic maps. The ..."
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Cited by 53 (20 self)
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We give the first example of a connected 4regular graph whose Laplace operator’s spectrum is a Cantor set, as well as several other computations of spectra following a common “finite approximation” method. These spectra are simple transforms of the Julia sets associated to some quadratic maps. The graphs involved are Schreier graphs of fractal groups of intermediate growth, and are also “substitutional graphs”. We also formulate our results in terms of Hecke type operators related to some irreducible quasiregular representations of fractal groups and in terms of the Markovian operator associated to noncommutative dynamical systems via which these fractal groups were originally defined in [Gri80]. In the computations we performed, the selfsimilarity of the groups is reflected in the selfsimilarity of some operators; they are approximated by finite counterparts whose spectrum is computed by an ad hoc factorization process.
On Asymptotic Cones and QuasiIsometry Classes of Fundamental Groups of 3Manifolds
 GAFA
, 1995
"... this paper was written. We are grateful to Richard Schwartz and Martin Bridson for remarks concerning the original manuscript of this paper. 2 Preliminaries ..."
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Cited by 53 (16 self)
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this paper was written. We are grateful to Richard Schwartz and Martin Bridson for remarks concerning the original manuscript of this paper. 2 Preliminaries
Quasiisometries and rigidity of solvable groups
, 2005
"... Abstract. In this note, we announce the first results on quasiisometric rigidity of nonnilpotent polycyclic groups. In particular, we prove that any group quasiisometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasiis ..."
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Cited by 42 (4 self)
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Abstract. In this note, we announce the first results on quasiisometric rigidity of nonnilpotent polycyclic groups. In particular, we prove that any group quasiisometric to the three dimenionsional solvable Lie group Sol is virtually a lattice in Sol. We prove analogous results for groups quasiisometric to R⋉R n where the semidirect product is defined by a diagonalizable matrix of determinant one with no eigenvalues on the unit circle. Our approach to these problems is to first classify all self quasiisometries of the solvable Lie group. Our classification of self quasiisometries for R⋉R n proves a conjecture made by Farb and Mosher in [FM3]. Our techniques for studying quasiisometries extend to some other classes of groups and spaces. In particular, we characterize groups quasiisometric to any lamplighter group, answering a question of de la Harpe [dlH]. Also, we prove that certain DiestelLeader graphs are not quasiisometric to any finitely generated group, verifying a conjecture of Diestel and Leader from [DL] and answering a question of Woess from [SW, Wo1]. We also prove that certain nonunimodular, nonhyperbolic solvable Lie groups are not quasiisometric to finitely generated groups. The results in this paper are contributions to Gromov’s program for classifying finitely generated groups up to quasiisometry [Gr2]. We introduce a new technique for studying quasiisometries, which we refer to as coarse differentiation.
Quasiisometries preserve the geometric decomposition of Haken manifolds
, 1995
"... . We prove quasiisometry invariance of the canonical decomposition for fundamental groups of Haken 3manifolds with zero Euler characteristic. We show that groups quasiisometric to Haken manifold groups with nontrivial canonical decomposition are finite extensions of Haken orbifold groups. As a by ..."
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Cited by 40 (7 self)
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. We prove quasiisometry invariance of the canonical decomposition for fundamental groups of Haken 3manifolds with zero Euler characteristic. We show that groups quasiisometric to Haken manifold groups with nontrivial canonical decomposition are finite extensions of Haken orbifold groups. As a byproduct we describe all 2dimensional quasiflats in the universal covers of nongeometric Haken manifolds. 1 Contents 1 Introduction 2 2 Preliminaries 4 2.1 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 3manifolds and their canonical decomposition . . . . . . . . . . . . . 5 2.3 Ultralimits and asymptotic cones . . . . . . . . . . . . . . . . . . . . 6 2.4 Busemann functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.5 Quasiisometric embeddings into piecewise Euclidean spaces . . . . . 7 3 Asymptotic cones of universal covers of Haken manifolds 8 3.1 Geometric components . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2 Separa...
Coarse differentiation of quasiisometries I: spaces not quasiisometric to Cayley graphs
, 2007
"... ..."
Dimension and rank for mapping class groups
, 2007
"... We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An application is a proof of BrockFarb’s Rank Conjecture wh ..."
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Cited by 34 (5 self)
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We study the large scale geometry of the mapping class group, MCG. Our main result is that for any asymptotic cone of MCG, the maximal dimension of locally compact subsets coincides with the maximal rank of free abelian subgroups of MCG. An application is a proof of BrockFarb’s Rank Conjecture which asserts that MCG has quasiflats of dimension N if and only if it has a rank N free abelian subgroup. (Hamenstadt has also given a proof of this conjecture, using different methods.) We also compute the maximum dimension of quasiflats in Teichmüller space with the WeilPetersson metric.
On the rigidity of discrete isometry groups of negatively curved spaces
 COMMENTARII MATHEMATICI HELVETICI
, 1997
"... We prove an ergodic rigidity theorem for discrete isometry groups of CAT(−1) spaces. We give explicit examples of divergence isometry groups with infinite covolume in the case of trees, piecewise hyperbolic 2polyhedra, hyperbolic BruhatTits buildings and rank one symmetric spaces. We prove that t ..."
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Cited by 21 (3 self)
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We prove an ergodic rigidity theorem for discrete isometry groups of CAT(−1) spaces. We give explicit examples of divergence isometry groups with infinite covolume in the case of trees, piecewise hyperbolic 2polyhedra, hyperbolic BruhatTits buildings and rank one symmetric spaces. We prove that two negatively curved Riemannian metrics, with conical singularities of angles at least 2π, on a closed surface, with boundary map absolutely continuous with respect to the PattersonSullivan measures, are isometric. For that, we generalize J.P. Otal’s result to prove that a negatively curved Riemannian metric, with conical singularities of angles at least 2π, on a closed surface, is determined, up to isometry, by its marked length spectrum.