Related papers: Free commuting involutions on closed two-dimension…

We prove that the minimal nontrivial finite quotient group of the mapping class group M_g of a closed orientable surface of genus g is the symplectic group PSp(2g,Z_2), for g = 3 and 4 (this might remain true, however, for arbitrary genus g…

A meander can be seen as a pair of transversally intersecting simple closed curves on a 2-sphere. We consider pairs of transversally intersecting simple closed curves on a closed oriented surface of arbitrary genus g. The number of such…

We study the combinatorial geometry of a random closed multicurve on a surface of large genus and of a random square-tiled surface of large genus. We prove that primitive components of a random multicurve represent linearly independent…

We will say that a group G possesses the Magnus property if for any two elements u,v in G with the same normal closure, u is conjugate to v or v^{-1}. We prove that some one-relator groups, including the fundamental groups of closed…

We study the configuration space of distinct, unordered points on compact orientable surfaces of genus $g$, denoted $S_g$. Specifically, we address the section problem, which concerns the addition of $n$ distinct points to an existing…

Let $S_g$ be the closed oriented surface of genus g and let $\text{Mod}(S_g)$ be the mapping class group. When the genus is at least 3, $\text{Mod}(S_g)$ can be generated by torsion elements. We prove the follow results. For $g \geq 4$,…

We study the configurations of genus 2 curves on the Fano surfaces of cubic threefolds. We establish a link between some involutive automorphisms acting on such a surface S and genus 2 curves on S. We give a partial classification of the…

A continuous action of a finite group $G$ on a closed orientable surface $X$ is said to be gpnf (Gilman purely non-free) if every element of $G$ has a fixed point on $X$. We prove that the biggest order {$\mu(g)$}, of a gpnf-action on a…

The moduli space $\M_{g,n}$ of $n-$pointed stable curves of genus $g$ is stratified by the topological type of the curves being parametrized: the closure of the locus of curves with $k$ nodes has codimension $k$. The one dimensional…

We prove that for every nonnegative integer $g$, there exists a bound on the number of ends of a complete, embedded minimal surface $M$ in $\mathbb{R}^3$ of genus $g$ and finite topology. This bound on the finite number of ends when $M$ has…

We use variational methods to construct a free boundary minimal surface in the three-dimensional unit ball with genus one, two boundary components and prismatic symmetry. Key ingredients are an extension of the equivariant min-max theory to…

We define a new integer invariant of a finite graph G, the freeness index, that measures the extent to which G can be embedded in the 3-sphere so that it and its subgraphs have ``simple" complements, i.e., complements which are homeomorphic…

Let $M$ be a simple manifold, and $F$ be a component of $\partial M$ of genus two. For a slope $\gamma$ on $F$, we denote by $M(\gamma)$ the manifold obtained by attaching a 2-handle to $M$ along a regular neighborhood of $\gamma$ on $F$.…

We show that the problem of deciding whether a knot in a fixed closed orientable 3-dimensional manifold bounds a surface of genus at most $g$ is in co-NP. This answers a question of Agol, Hass, and Thurston in 2002. Previously, this was…

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of…

We extend the modular orbits method of constructing a two-dimensional orbifold conformal field theory to higher genus Riemann surfaces. We find that partition functions on surfaces of arbitrary genus can be constructed by a straightforward…

Given a Riemannian surface, we consider a naturally embedded graph which captures part of the topology and geometry of the surface. By studying this graph, we obtain results in three different directions. First, we find bounds on the…

We find the minimum dilatation of pseudo-Anosov homeomorphisms that stabilize an orientable foliation on surfaces of genus three, four, or five, and provide a lower bound for genus six to eight. Our technique also simplifies Cho and Ham's…

Let $M$ be a closed Riemannian surface of genus $g$. We construct a family of 1-cycles on $M$ that represents a non-trivial element of the k'th homology group of the space of cycles and such that the mass of each cycle is bounded above by…

We describe smooth rational projective algebraic surfaces over an algebraically closed field of characteristic different from 2 which contain $n \ge \b_2-2$ disjoint smooth rational curves with self-intersection -2, where $\b_2$ is the…