Geodesics in Margulis spacetimes

Goldman, William M.; Labourie, François

doi:10.1017/s0143385711000678

Cited by 9 publications

(15 citation statements)

References 20 publications

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“…A complete geodesic on a quotient space of a hyperbolic space or a Lorentzian space is nonwandering if it is bounded in both directions, or equivalently, the closure of the forward part is compact and so is that of the backward part, or the α-limit and the ω-limit are both nonempty and compact. This is the same as the term "recurrent" in [45] and [18], which does not agree with the common usage in the dynamical system theory whereas the recurrence set they discuss is the same as the nonwandering set as in Eberlein [35] and Katok and Hasselblatt [50]. Both endpoints of the geodesics lie in the limit sets and they comprise the nonwandering set by Corollary 3.8 in [35].…”

Section: Denote Bymentioning

confidence: 90%

“…(See §3.6.1.) The uniform bounds on h n follow from dynamical properties established in [45] as stated by equation (19).…”

Section: γ Acts Properly On E ∪σmentioning

confidence: 99%

“…Charette, Drumm, Goldman, and Labourie [31], [33], [34] made contributions to understanding such spacetimes with parallel Lorentzian structures culminating in recent work [46] and [45]. The complete classification of such spacetimes is currently going on with the work of Charette, Drumm, Goldman, Labourie, Margulis and Minsky [15], [16], [46], and [47].…”

Section: Introductionmentioning

confidence: 99%

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Topological tameness of Margulis Spacetimes

Choi¹,

Goldman²

2017

American Journal of Mathematics

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Abstract. We show that Margulis spacetimes without parabolic holonomy elements are topologically tame. A Margulis spacetime is the quotient of the 3-dimensional Minkowski space by a free proper isometric action of the free group of rank ≥ 2. We will use our particular point of view that the Margulis spacetime is a manifold-with-boundary with an RP 3 -structure in an essential way. The basic tools are a bordification by a closed RP 2 -surface with free holonomy group, and the work of Goldman, Labourie, and Margulis on geodesics in the Margulis spacetimes and 3-manifold topology.

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Section: Denote Bymentioning

confidence: 90%

“…(See §3.6.1.) The uniform bounds on h n follow from dynamical properties established in [45] as stated by equation (19).…”

Section: γ Acts Properly On E ∪σmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Topological tameness of Margulis Spacetimes

Choi¹,

Goldman²

2017

American Journal of Mathematics

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“…Hence −f − is also Livšic cohomologous to a strictly positive function. Then using Lemma 7 of [GL12] we get that there exists c > 0 such that the function…”

Section: Existence Of Proper Actionsmentioning

confidence: 99%

Avatars of Margulis invariants and proper actions

Ghosh¹

2018

Preprint

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In this article, we interpret affine Anosov representations of any word hyperbolic group in SO0(n − 1, n) ⋉ R 2n−1 as infinitesimal versions of representations of word hyperbolic groups in SO0(n, n) which are both Anosov in SO0(n, n) with respect to the stabilizer of an oriented (n−1)-dimensional isotropic plane and Anosov in SL(2n, R) with respect to the stabilizer of an oriented n-dimensional plane. Moreover, we show that representations of word hyperbolic groups in SO0(n, n) which are Anosov in SO0(n, n) with respect to the stabilizer of an oriented (n − 1)dimensional isotropic plane, are Anosov in SL(2n, R) with respect to the stabilizer of an oriented n-dimensional plane if and only if its action on SO0(n, n)/SO0(n − 1, n) is proper. In the process, we also provide various different interpretations of the Margulis invariant.

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“…They were originally discovered by Margulis [59] as counterexamples to a question of Milnor. Ghosh [32] used work of Goldman, Labourie and Margulis [34,35] to interpret holonomy maps of Margulis space times (without cusps) as "Anosov representations" into the (non-semisimple) Lie group Aff(R 3 ) of affine automorphisms of R 3 . Ghosh [33] was then able to adapt the techniques of [17] to produce a pressure form on the analytic manifold M of (conjugacy classes of) holonomy maps of Margulis space times of fixed rank (with no cusps).…”

Section: Generalizations and Consequencesmentioning

confidence: 99%

An introduction to pressure metrics for higher Teichmüller spaces

Bridgeman

Canary

Sambarino

2017

Ergod. Th. Dynam. Sys.

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We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichmüller spaces. Our higher Teichmüller spaces will be spaces of Anosov representations of a word-hyperbolic group into a semi-simple Lie group. We begin by discussing our construction in the classical setting of the Teichmüller space of a closed orientable surface of genus at least 2, then we explain the construction for Hitchin components and finally we treat the general case. This paper surveys results of Bridgeman, Canary, Labourie and Sambarino, The pressure metric for Anosov representations, and discusses questions and open problems which arise.

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Geodesics in Margulis spacetimes

Cited by 9 publications

References 20 publications

Topological tameness of Margulis Spacetimes

Topological tameness of Margulis Spacetimes

Avatars of Margulis invariants and proper actions

An introduction to pressure metrics for higher Teichmüller spaces

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