Dan Margalit
Georgia Institute of Technology
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Archive | 2011
Benson Farb; Dan Margalit
Given a compact connected orientable surface S there are two fundamental objects attached: a group and a space. The group is the mapping class group of S, denoted by Mod(S). This group is defined by the isotopy classes of orientation-preserving homeomorphism from S to itself. Equivalently, Mod(S) may be defined using diffeomorphisms instead of homeomorphisms or homotopy classes instead of isotopy classes. The space is the Teichmüller space of S, Teich(S). Teichmüller space and moduli space are fundamental objects in fields like low-dimensional topology, algebraic geometry and mathematical physics. If X (S) < 0, the Teichmüller space can be thought of as the set of homotopy classes of hyperbolic structures of S or, equivalently, as the set of isotopy classes of hyperbolic metrics on S, HypMet(S). The group and the space are connected through the moduli space in the following way. The group of orientation-preserving diffeomorphisms of S, Diff+(S) acts on HypMet(S) and this action descends to an action of Mod(S) on Teich(S) which is properly discontinuous. The quotient space,
Geometry & Topology | 2004
Tara E. Brendle; Dan Margalit
Let K be the subgroup of the extended mapping class group, Mod(S), generated by Dehn twists about separating curves. Assuming that S is a closed, orientable surface of genus at least 4, we confirm a conjecture of Farb that Comm(K) � Aut(K) � Mod(S). More generally, we show that any injection of a finite index subgroup of K into the Torelli group I of S is induced by a homeomorphism. In particular, this proves that K is co-Hopfian and is characteristic in I. Further, we recover the result of Farb and Ivanov that any injection of a finite index subgroup of I into I is induced by a homeomorphism. Our method is to reformulate these group theoretic statements in terms of maps of curve complexes.
Commentarii Mathematici Helvetici | 2007
Robert W. Bell; Dan Margalit
We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a sphere with at least 5 punctures. In particular, we show that any injective homomorphism between these groups is given by a homeomorphism of a punctured sphere together with a map to the integers. The technique, following Ivanov, is to prove that every superinjective map of the curve complex of a sphere with at least 5 punctures is induced by a homeomorphism. We also determine the automorphism group of the pure braid group on at least 4 strands.
arXiv: Differential Geometry | 2007
Jeffrey F. Brock; Dan Margalit
We extend a theorem of Masur and Wolf which states that given a finite area hyperbolic surface S, every isometry of the Teichmuller space for S with the Weil-Petersson metric is induced by an element of the mapping class group for S. Our argument handles the previously untreated cases of the four times-punctured sphere, the once-punctured torus, and the twice-punctured torus.
American Journal of Mathematics | 2008
Benson Farb; Christopher J. Leininger; Dan Margalit
The theme of this paper is that algebraic complexity implies dynamical complexity for pseudo-Anosov homeomorphisms of a closed surface
Inventiones Mathematicae | 2007
Mladen Bestvina; Kai-Uwe Bux; Dan Margalit
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Inventiones Mathematicae | 2015
Tara E. Brendle; Dan Margalit; Andrew Putman
of genus
Algebraic & Geometric Topology | 2002
Dan Margalit
g
Geometry & Topology | 2009
Dan Margalit; Saul Schleimer
. Penner proved that the logarithm of the minimal dilatation for a pseudo-Anosov homeomorphism of
arXiv: Geometric Topology | 2012
Allen Hatcher; Dan Margalit
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