Related papers: A finite presentation for the hyperelliptic mappin…

Motivated by the Lawrence-Krammer-Bigelow representations of the classical braid groups, we study the homology of unordered configurations in an orientable genus-$g$ surface with one boundary component, over non-commutative local systems…

We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping…

The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. What makes this study tractable is that there is a surjective map from the hyperelliptic mapping class group to a mapping…

This paper has 3 principal goals: (1) to survey what is know about mapping class and Torelli groups of simply connected compact Kaehler manifolds, (2) supplement these results, and (3) present a list of questions and open problems to…

Let M be a closed n-dimensional manifold, n > 2, whose first real cohomology group H 1 (M ; R) is non-zero. We present a general method for constructing a Morse 1-form $\alpha$ on M , closed but non-exact, and a pseudo-gradient X such that…

A crosscap transposition is an element of the mapping class group of a nonorientable surface represented by a homeomorphism supported on a one-holed Klein bottle and swapping two crosscaps. We prove that the mapping class group of a compact…

Motivated by problems arising in the complex analysis of perturbative quantum field theory, we investigate the homology of finite unions of certain non-degenerate quadratic affine hypersurfaces of complex dimension $n$ in general position.…

Let $\Gamma$ be either the mapping class group of a closed surface of genus $\geq 2$, or the automorphism group of a free group of rank $\geq 3$. Given any homological representation $\rho$ of $\Gamma$ corresponding to a finite cover, and…

Let $\Sigma_{g,n}$ be an orientable surface of genus $g$ with $n$ punctures. We study actions of the mapping class group of $\Sigma_{g,n}$ via Hodge-theoretic and arithmetic techniques. We show that if $$\rho: \pi_1(\Sigma_{g,n})\to…

We derive an obstruction to representing a homology class of a symplectic 4-manifold by an embedded, possibly disconnected, symplectic surface.

Let M be a surface (possibly nonorientable) with punctures and/or boundary components. The paper is a study of ``geometric subgroups'' of the mapping class group of M, that is subgroups corresponding to inclusions of subsurfaces (possibly…

This paper studies the set of finite groups appearing as $\pi_1(M)/\pi_1(M)^{(n)}$, where $M$ is a closed, orientable 3-manifold and $\pi_1(M)^{(n)}$ denotes the $n$-th term of the derived series of $\pi_1(M)$. Our main result is that if…

Flat surfaces that correspond to meromorphic $1$-forms or to meromorphic quadratic differentials containing poles of order two and higher are surfaces of infinite area. We classify groups that appear as Veech groups of translation surfaces…

We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed…

Let $(S,\, \ast)$ be a closed oriented surface with a marked point, let $G$ be a fixed group, and let $\rho\colon\pi_1(S) \longrightarrow G$ be a representation such that the orbit of $\rho$ under the action of the mapping class group…

One of the consequences of residual finiteness of triangle groups is that for any given hyperbolic triple $(\ell,m,n)$ there exist infinitely many regular hypermaps of type $(\ell,m,n)$ on compact orientable surfaces. The same conclusion…

We derive two types of linearity conditions for mapping class groups of orientable surfaces: one for once-punctured surface, and the other for closed surface, respectively. For the once-punctured case, the condition is described in terms of…

Let $\mathcal{N}_g$ be the mapping class group of a non-orientable closed surface. We prove that the proper cohomological dimension, the proper geometric dimension, and the virtual cohomological dimension of $\mathcal{N}_g$ are equal…

We prove that the hyperelliptic mapping class group of a nonorientable surface of genus $g\geq 4$ has a faithful linear representation of dimension $g^2-1$ over $\mathbb{R}$.

We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.