Dan Margalit

A primer on mapping class groups

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,

Commensurations of the Johnson kernel.

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.

Injections of Artin groups

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.

WEIL-PETERSSON ISOMETRIES VIA THE PANTS COMPLEX

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.

The lower central series and pseudo-Anosov dilatations

The theme of this paper is that algebraic complexity implies dynamical complexity for pseudo-Anosov homeomorphisms of a closed surface

Dimension of the Torelli group for Out(Fn)

S_g

Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t = 1

of genus

A lantern lemma

g

Dehn twists have roots

. Penner proved that the logarithm of the minimal dilatation for a pseudo-Anosov homeomorphism of

Generating the Torelli group

S_g