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An Introduction to Geometric Topology
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This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds. It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphisms of surfaces (via Bonahon's geodesic currents), the prime and JSJ decomposition, the topological and geometric classification of Seifert manifolds, and Thurston's hyperbolic Dehn filling Theorem.
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Cited by 7 Pith papers
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Normalizers of lattices and isometry groups of arithmetic hyperbolic manifolds
Torsion-free arithmetic lattices in PSL(2,C) and simplest-type lattices in PO(n,1) each normalize arbitrarily many sublattices, and every finite group is the full isometry group of infinitely many arithmetic hyperboli...
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Behaviour of the Schwarzian derivative on long complex projective tubes
On long complex projective tubes, the Schwarzian derivative is asymptotically (1/2z^2)(1+4π^2/ℓ^2) dz^2 up to exponentially small errors, giving control of renormalized volume under earthquakes and grafting.
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Genericity of hyperbolic 3-manifolds via Dehn surgery
A counting measure on braids and Dehn fillings shows that generic links and generic closed orientable 3-manifolds are hyperbolic.
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An EZ-structure for the mapping class group
Hamenstadt defines an explicit geometric boundary X(S) for Mod(S) and compactifies the thick part of Teichmuller space by it, producing an EZ-structure whose boundary has strong dynamical properties.
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On dynamics of the Mapping class group action on relative $\text{PSL}(2,\mathbb{R})$-Character Varieties
Simple-stable representations of punctured surface groups into PSL(2,R) form a domain of discontinuity for the mapping class group action, and admissible hyperbolic cone surface holonomies are simple-stable, with one-...
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On the Dax invariants of $S^2$-bundles over surfaces
Relative Dax invariants classify dual-paired embeddings of Σ into the nontrivial S^{2}-bundle Σ ⋉ S^{2} up to isotopy and yield a surjective homomorphism MCG(Σ ⋉ S^{2}) → ℤ^∞.
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Volumes of end-periodic mapping tori
An expository paper presenting theorems that bound volumes of end-periodic mapping tori by pants-graph translation length, an infinite-type analogue of Brock's theorem.
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