Related papers: Addendum to: Commensurations of the Johnson kernel

In a classic paper, Gerstenhaber showed that first order deformations of an associative k-algebra A are controlled by the second Hochschild cohomology group of A. More generally, any n-parameter first order deformation of A gives, due to…

We characterize convex cocompact subgroups of mapping class groups that arise as subgroups of specially embedded right-angled Artin groups. That is, if the right-angled Artin group G in Mod(S) satisfies certain conditions that imply G is…

We prove that any geometrically connected curve $X$ over a field $k$ is an algebraic $K(\pi,1)$, as soon as its geometric irreducible components have nonzero genus. This means that the cohomology of any locally constant constructible…

We show that one can find two nonisomorphic curves over a field K that become isomorphic to one another over two finite extensions of K whose degrees over K are coprime to one another. More specifically, let K_0 be an arbitrary prime field…

Let $S$ be a compact Riemann surface and $G$ a group of conformal automorphisms of $S$ with $S_0 = S/G$. $S$ is a finite regular branched cover of $S_0$. If $U$ denotes the unit disc, let $\Gamma$ and $\Gamma_0$ be the Fuchsian groups with…

In this thesis, we prove several results concerning field-theoretic invariants of knots and 3-manifolds. In Chapter 2, for any knot $K$ in a closed, oriented 3-manifold $M$, we use $SU(2)$ representation spaces and the Lagrangian field…

Let G be a finite group and let k be a field of characteristic p. Given a finitely generated indecomposable non-projective kG-module M, we conjecture that if the Tate cohomology $\HHHH^*(G, M)$ of G with coefficients in M is finitely…

Let a and b be two simple closed curves on an orientable surface S such that their geometric intersection number is greater than 1. It is known that the group generated by corresponding Dehn twists t_a and t_b is isomorphic to the free…

A theorem of Farb and Handel asserts that for $N\ge 4$, the natural inclusion from $\mathrm{Out}(F_N)$ into its abstract commensurator is an isomorphism. We give a new proof of their result, which enables us to generalize it to the case…

Let K be a knot in S^3 of genus g and let n>0. We show that if rk HFK(K,g) < 2^{n+1} (where HFK denotes knot Floer homology), in particular if K is an alternating knot such that the leading coefficient a_g of its Alexander polynomial…

Let $G$ be a finite group acting on a connected compact surface $\Sigma$, and $M$ be an integer homology 3-sphere. We show that if each element of $G$ is extendable over $M$ with respect to a fixed embedding $\Sigma\rightarrow M$, then $G$…

Let $A$ be a simple abelian surface over an algebraically closed field $k$. Let $S\subset A(k)$ be the set of torsion points $x$ of $A$ such that there exists a genus $2$ curve $C$ and a map $f: C\to A$ such that $x$ is in the image of $f$,…

The Jones-Witten theory gives rise to representations of the (extended) mapping class group of any closed surface Y indexed by a semi-simple Lie group G and a level k. In the case G=SU(2) these representations (denoted V_A(Y)) have a…

We prove that the $k$th term of the Johnson filtration of a closed, orientable surface of genus $g \geq 2$ has cohomological dimension $2g - 3$ for all $k \geq 3$ and $g \geq 2$. This answers a question of Farb and Bestvina--Bux--Margalit.

Let $S_{g,n}$ be a closed oriented hyperbolic surface of genus $g$ with $n$ marked points, with the understanding that $S_{g,0}=S_g$. Let $\mathrm{Mod}(S_{h,n})$ be the mapping class group of $S_{h,n}$ and $\mathrm{LMod}_p(S_{h,n})$ be the…

In this paper we prove that finite index subgroups of genus 3 mapping class and Torelli groups that contain the group generated by Dehn twists on bounding simple closed curves are not Kahler. These results are deduced from explicit…

Given a compact Riemann surface X with an action of a finite group G, the group algebra Q[G] provides an isogenous decomposition of its Jacobian variety JX, known as the group algebra decomposition of JX. We obtain a method to concretely…

In the first part of this paper we prove that the mapping class subgroups generated by the $D$-th powers of Dehn twists (with $D\geq 2$) along a sparse collection of simple closed curves on an orientable surface are right angled Artin…

Consider a connected orientable surface $S$ of infinite topological type, i.e. with infinitely-generated fundamental group. We describe the large-scale geometry of arbitrary connected subgraphs of the arc complex $A(S)$ and curve complex…

Let K be the kernel of an epimorphism G -> Z, where G is a finitely presented group. If K has infinitely many subgroups of index 2, 3, or 4, then it has uncountably many. Moreover, if K is the commutator subgroup of a classical knot group…