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72
Right-veering diffeomorphisms of compact surfaces with boundary II
- INVENT MATH. 169
, 2008
"... We continue our study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary, introduced in [HKM2]. We conduct a detailed study of the case when the surface is a punctured torus; in particular, we exhibit the difference between the monoid of right-veering ..."
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Cited by 46 (4 self)
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We continue our study of the monoid of right-veering diffeomorphisms on a compact oriented surface with nonempty boundary, introduced in [HKM2]. We conduct a detailed study of the case when the surface is a punctured torus; in particular, we exhibit the difference between the monoid of right-veering diffeomorphisms and the monoid of products of positive Dehn twists, with the help of the Rademacher function. We then generalize to the braid group Bn on n strands by relating the signature and the Maslov index. Finally, we discuss the symplectic fillability in the pseudo-Anosov case by comparing with the work of Roberts [Ro1, Ro2].
Mapping class groups and moduli spaces of curves
- Proc. Symposia in Pure Math
, 1997
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The modular group action on real SL(2)-characters of a one-holed torus
, 2003
"... The group Γ of automorphisms of the polynomial is isomorphic to κ(x, y, z) = x 2 + y 2 + z 2 − xyz − 2 PGL(2, Z) ⋉ (Z/2 ⊕ Z/2). For t ∈ R, the Γ-action on κ −1 (t) ∩ R 3 displays rich and varied dynamics. The Γ-action preserves a Poisson structure defining a Γ-invariant area form on each κ −1 (t ..."
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Cited by 33 (0 self)
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The group Γ of automorphisms of the polynomial is isomorphic to κ(x, y, z) = x 2 + y 2 + z 2 − xyz − 2 PGL(2, Z) ⋉ (Z/2 ⊕ Z/2). For t ∈ R, the Γ-action on κ −1 (t) ∩ R 3 displays rich and varied dynamics. The Γ-action preserves a Poisson structure defining a Γ-invariant area form on each κ −1 (t) ∩ R 3. For t < 2, the action of Γ is properly discontinuous on the four contractible components of κ −1 (t) ∩ R 3 and ergodic on the compact component (which is empty if t < −2). The contractible components correspond to Teichmüller spaces of (possibly singular) hyperbolic structures on a torus ¯ M. For t = 2, the level set κ−1 (t) ∩ R3 consists of characters of reducible representations and comprises two ergodic components corresponding to actions of GL(2, Z) on (R/Z) 2 and R 2 respectively. For 2 < t ≤ 18, the action of Γ on κ −1 (t) ∩ R 3 is ergodic. Corresponding to the Fricke space of a three-holed sphere is a Γ-invariant open subset Ω ⊂ R 3 whose components are permuted freely by a subgroup of index 6 in Γ. The level set κ −1 (t) ∩ R 3 intersects Ω if and only if t> 18, in which case the Γ-action on the complement (κ −1 (t) ∩ R 3) − Ω is ergodic.
An algorithm for simple curves on surfaces
- J. London Math. Soc
, 1984
"... Let M be a compact orientable surface with non-empty boundary and with X{M) < 0, and let T = nlM. Let C be the free homotopy class of a closed loop on M and let W = W{C) be a word in a fixed set of generators T which represents C. In this paper we give an algorithm to decide, starting with W, whe ..."
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Cited by 22 (1 self)
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Let M be a compact orientable surface with non-empty boundary and with X{M) < 0, and let T = nlM. Let C be the free homotopy class of a closed loop on M and let W = W{C) be a word in a fixed set of generators T which represents C. In this paper we give an algorithm to decide, starting with W, whether C has a simple
Solvable Fundamental Groups of compact 3-manifolds
- TRANS. AMER. MATH. SOC
, 1972
"... A classification is given for groups which can occur as the fundamental group of some compact 3-manifold. In most cases we are able to determine the topological structure of a compact 3-manifold whose fundamental group is known to be solvable. Using the results obtained for solvable groups, we are ..."
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Cited by 21 (0 self)
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A classification is given for groups which can occur as the fundamental group of some compact 3-manifold. In most cases we are able to determine the topological structure of a compact 3-manifold whose fundamental group is known to be solvable. Using the results obtained for solvable groups, we are able to extend some known results concerning nilpotent groups of closed 3-manifolds to the more general class of compact 3-manifolds. In the final section it is shown that each nonfinitely generated abelian group which occurs as a subgroup of the fundamental group of a 3-manifold is a subgroup of the additive group of rationals.
SYMBOLIC DYNAMICS FOR THE MODULAR SURFACE AND BEYOND
, 2007
"... In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording ..."
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Cited by 20 (2 self)
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In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated Fuchsian groups. Another method, of arithmetic nature, uses continued fraction expansions of the end points of the geodesic at infinity and is even older—it comes from the Gauss reduction theory. Introduced to dynamics by E. Artin in a 1924 paper, this method was used to exhibit dense geodesics on the modular surface. For 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. We present some of the ideas, results (old and recent), and interpretations that illustrate the multiple facets of the subject.
CLASSIFYING SPACES FOR PROPER ACTIONS OF MAPPING CLASS GROUPS
, 905
"... Abstract. 1 We describe a construction of cocompact models for the classifying spaces EΓ s g,r, where Γ s g,r stands for the mapping class group of an oriented surface of genus g with r boundary components and s punctures. Our construction uses a cocompact model for EΓ 0 g,0 as an input, a case whic ..."
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Cited by 12 (0 self)
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Abstract. 1 We describe a construction of cocompact models for the classifying spaces EΓ s g,r, where Γ s g,r stands for the mapping class group of an oriented surface of genus g with r boundary components and s punctures. Our construction uses a cocompact model for EΓ 0 g,0 as an input, a case which has been dealt with in [3]. We then proceed by induction on r and s. 1.
The Kazhdan Property of the Mapping Class Group of Closed Surfaces and the First Cohomology Group of Their Cofinite Subgroups
- DEPARTMENT OF MATHEMATICS, MIDDLE EAST TECHNICAL UNIVERSITY
"... In the following we show that the mapping class group of a closed surface of genus 2 does not satisfy the Kazhdan property by constructing subgroups of finite index having a non-vanishing first cohomology group. We also construct some subgroups of finite index in the mapping class group of a genus 3 ..."
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Cited by 9 (0 self)
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In the following we show that the mapping class group of a closed surface of genus 2 does not satisfy the Kazhdan property by constructing subgroups of finite index having a non-vanishing first cohomology group. We also construct some subgroups of finite index in the mapping class group of a genus 3 surface and calculate their first cohomology groups, which all turn out to be trivial. Most of the calculations have been carried out by the aid of a computer using the programming language GAP (see [Sc]).
