ℝ-trees and laminations for free groups I: algebraic laminations

Coulbois, Thierry; Hilion, Arnaud; Lustig, Martin

doi:10.1112/jlms/jdn052

Cited by 54 publications

(71 citation statements)

References 11 publications

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“…Here supp. / is the support of and L 2 .T / is the dual algebraic lamination of T (see Coulbois, Hilion and Lustig [10]). That result in turn is applied in [27] to the notions of a filling conjugacy class and a filling current as well as to obtain results about bounded translation equivalence in F N .…”

Section: Introductionmentioning

confidence: 99%

Geometric intersection number and analogues of the curve complex for free groups

Kapovich

Lustig²

2009

Geom. Topol.

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144

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1805Geometric intersection number and analogues of the curve complex for free groups ILYA KAPOVICH MARTIN LUSTIGFor the free group F N of finite rank N 2 we construct a canonical Bonahon-type, continuous and Out.F N /-invariant geometric intersection formHere cv.F N / is the closure of unprojectivized Culler-Vogtmann Outer space cv.F N / in the equivariant Gromov-Hausdorff convergence topology (or, equivalently, in the length function topology). It is known that cv.F N / consists of all very small minimal isometric actions of F N on R-trees. The projectivization of cv.F N / provides a free group analogue of Thurston's compactification of Teichmüller space.As an application, using the intersection graph determined by the intersection form, we show that several natural analogues of the curve complex in the free group context have infinite diameter.

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Section: Introductionmentioning

confidence: 99%

Geometric intersection number and analogues of the curve complex for free groups

Kapovich

Lustig²

2009

Geom. Topol.

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144

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“…The analogue to a minimal symbolic dynamical system is then given by algebraic laminations. An attractive algebraic lamination of an automorphism is a set of geodesic lines in the free group which is closed (for the topology induced by the boundary topology), invariant under the action of the group and flip-invariant (i.e., orientation-invariant); therefore it is the analog to a substitutive dynamical system [58]. However, explicitly building such an attractive algebraic lamination is far from trivial; indeed, the constructions used for substitution cannot be used since iterations of automorphisms of free groups produce cancelations so that infinite fixed words cannot be generated easily.…”

Section: Invariants In Dynamics and Geometrymentioning

confidence: 99%

Topological properties of Rauzy fractals

Siegel¹,

Thuswaldner²

2009

Mémoires Soc. Math. France

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Abstract. -Substitutions are combinatorial objects (one replaces a letter by a word) which produce sequences by iteration. They occur in many mathematical fields, roughly as soon as a repetitive process appears. In the present monograph we deal with topological and geometric properties of substitutions, i.e., we study properties of the Rauzy fractals associated to substitutions.To be more precise, let σ be a substitution over the finite alphabet A. We assume that the incidence matrix of σ is primitive and its dominant eigenvalue is a unit Pisot number (i.e., an algebraic integer greater than one whose norm is equal to one and all of whose Galois conjugates are of modulus strictly smaller than one). It is well-known that one can attach to σ a set T which is now called central tile or Rauzy fractal of σ. Such a central tile is a compact set that is the closure of its interior and decomposes in a natural way in n = #A subtiles T (1), . . . , T (n). The central tile as well as its subtiles are graph directed self-affine sets that often have fractal boundary.Pisot substitutions and central tiles are of high relevance in several branches of mathematics like tiling theory, spectral theory, Diophantine approximation, the construction of discrete planes and quasicrystals as well as in connection with numeration like generalized continued fractions and radix representations. The questions coming up in all these domains can often be reformulated in terms of questions related to the topology and geometry of the underlying central tile.After a thorough survey of important properties of unit Pisot substitutions and their associated Rauzy fractals the present monograph is devoted to the investigation of a variety of topological properties of T and its subtiles. Our approach is an algorithmic one. In particular, we dwell upon the question whether T and its subtiles induce a tiling, calculate the Hausdorff dimension of their boundary, give criteria for their connectivity and homeomorphy to a disk and derive properties of their fundamental group.The basic tools for our criteria are several classes of graphs built from the description of the tiles T (i) (1 ≤ i ≤ n) as graph directed iterated function systems and from the structure of the tilings induced by these tiles. These graphs are of interest in their own right. For instance, they can be used to construct the boundaries ∂T as well as ∂T (i) (1 ≤ i ≤ n) and all points where two, three or four different tiles of the induced tilings meet.When working with central tiles in one of the above mentioned contexts it is often useful to know such intersection properties of tiles. In this sense the present monograph also aims at providing tools for "everyday's life" when dealing with topological and geometric properties of substitutions.Many examples are given throughout the text in order to illustrate our results. Moreover, we give perspectives for further directions of research related to the topics discussed in this monograph.

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“…We refer the reader to [CouHL1,2,3] for detailed background material on algebraic laminations in the context of free groups. We shall only state some basic definitions and facts here.…”

Section: Laminationsmentioning

confidence: 99%

“…Note that in [CouHL1,2,3] laminary languages are defined only with respect to a free basis of a free group. However, it is easy to see that the definition and the basic results listed here extend to an arbitrary simplicial chart, that is not necessarily a wedge of loop-edges.…”

Section: Definition 31 (Algebraic Laminations) An Algebraic Laminatmentioning

confidence: 99%

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Intersection Form, Laminations and Currents on Free Groups

Kapovich

Lustig

2009

Geom. Funct. Anal.

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Abstract. Let F be a free group of rank N ≥ 2, let µ be a geodesic current on F and let T be an R-tree with a very small isometric action of F . We prove that the geometric intersection number T, µ is equal to zero if and only if the support of µ is contained in the dual algebraic lamination L 2 (T ) of T . Applying this result, we obtain a generalization of a theorem of Francaviglia regarding length spectrum compactness for currents with full support. We use the main result to obtain "unique ergodicity" type properties for the attracting and repelling fixed points of atoroidal iwip elements of Out(F ) when acting both on the compactified outer Space and on the projectivized space of currents. We also show that the sum of the translation length functions of any two "sufficiently transverse" very small F -trees is bilipschitz equivalent to the translation length function of an interior point of the outer space. As another application, we define the notion of a filling element in F and prove that filling elements are "nearly generic" in F . We also apply our results to the notion of bounded translation equivalence in free groups.

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ℝ-trees and laminations for free groups I: algebraic laminations

Cited by 54 publications

References 11 publications

Geometric intersection number and analogues of the curve complex for free groups

Geometric intersection number and analogues of the curve complex for free groups

Topological properties of Rauzy fractals

Intersection Form, Laminations and Currents on Free Groups

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