Quasi-convexity and shrinkwrapping
type: Item
instance of: scholarly article
title: Quasi-convexity and shrinkwrapping
zbMATH Open document ID: 1184.57012
author: Hossein Namazi
published in: Algebraic \& Geometric Topology
publication date: 2010-01-12T00:00:00Z
review text: Let \(S\) be a closed surface of genus \(>1\), \(N\) be an oriented hyperbolic 3-manifold and \(j :S\rightarrow N\) be a \(\pi_1\)-injective map. Let us denote by \(C(B,j)\) the set of homotopy classes of noncontractible simple loops \(\alpha\) where \(j(\alpha)\) has a representative of length \(\leq B\) in \(N\). This is a subspace of the complex of curves \(C(S)\) of \(S\). The following theorem was proved in [\textit{Y. N. Minsky}, Invent. Math. 146, No.~1, 143--192 (2001; Zbl 1061.37026)]. Theorem. For \(B\) large enough depending on \(\chi(S)\), the set \(C(B,j)\) is \(K\)-quasi-convex in \(C(S)\), with \(K\) depending only on \(\chi(S)\) and \(B\). (this means that for \(p,q\in C(B,j)\) every geodesic connecting them is in the \(K\)-neighborhood of \(C(B,j)\)). The aim of the paper under review is to extend the preceding theorem. In particular the condition about \(\pi_1\)-injectivity of \(j\) is replaced by a weaker condition. Such a result is seen as a step to understand the following question, addressed in the paper under review: \textbf{Question} Do there exist constants \(B,K\) such that, for every irreducible representation \(\rho : \pi_1(S)\rightarrow \mathrm{PSL}_2(\mathbb{C})\), \(C(B,\rho)\) is \(K\)-quasi-convex in \(C(S)\)?
reviewed by: François Fillastre
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 30F40
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 5655953
zbMATH Keywords: complex of curves
zbMATH Keywords: quasi-convexity
zbMATH Keywords: shrinkwrapping
MaRDI profile type: Publication
full work available at URL: agt.2009.9.2443
OpenAlex ID: W2007027724
Wikidata QID: Q125360703
cites work: Markoff triples and quasifuchsian groups
cites work: The classification of Kleinian surface groups. II: The Ending lamination conjecture
cites work: Collars in Kleinian groups
cites work: Shrinkwrapping and the taming of hyperbolic 3-manifolds
cites work: A covering theorem for hyperbolic 3-manifolds and its applications
cites work: Least area incompressible surfaces in 3-manifolds
cites work: On the geometric and topological rigidity of hyperbolic 3-manifolds
cites work: Deformations of hyperbolic \(3\)-cone-manifolds
cites work: Geometry of the complex of curves. I: Hyperbolicity
cites work: Geometry of the complex of curves. II: Hierarchical structure
cites work: Q4662434
cites work: A Lower Bound for the Volume of Hyperbolic 3-Manifolds
cites work: The classification of Kleinian surface groups. I: Models and bounds
cites work: Bounded geometry for Kleinian groups.
cites work: Existence of ruled wrappings in hyperbolic 3-manifolds
cites work: On irreducible 3-manifolds which are sufficiently large
DOI: 10.2140/AGT.2009.9.2443
Recommended article: QUASICONVEXITY OF BANDS IN HYPERBOLIC 3-MANIFOLDS
Recommended article: Q4662434
Recommended article: Length bounds on curves arising from tight geodesics
Recommended article: Traces, lengths, and simplicity for loops on surfaces
Recommended article: Systems of bands in hyperbolic 3-manifolds
Recommended article: Surface groups in some surgered manifolds
Recommended article: Stability of quasi-geodesics in Teichmüller space
Recommended article: Some new behaviour in the deformation theory of Kleinian groups
Recommended article: Continuity of the bending map
Recommended article: Uniform quasiconvexity of the disc graphs in the curve graphs
description: scientific article; zbMATH DE number 5655953