Related papers: Generating the mapping class group of a punctured …

We prove new conditional bounds on the the $m$-torsion of class groups of number fields of any fixed degree, for $m=2$, $3$, $4$, and $5$. Our methods first recast the problem in the language of class groups of Galois modules, which allows…

An involution is a permutation that is its own inverse. Given a permutation $\sigma$ of $[n],$ let $\mathbf{N}_{n}(\sigma)$ denote the number of ways to write $\sigma$ as a product of two involutions of $[n].$ If we endow the symmetric…

Gay and Kirby recently introduced the concept of a trisection for arbitrary smooth, oriented closed 4-manifolds, and with it a new topological invariant, called the trisection genus. This paper improves and implements an algorithm due to…

Finding the number of maximal subgroups of infinite index of a finitely generated group is a natural problem that has been solved for several classes of `geometric' groups (linear groups, hyperbolic groups, mapping class groups, etc). Here…

In [3] is was shown that for any group $G$ whose rank (i.e., minimal number of generators) is at most 3, and any finite index subgroup $H\leq G$ with index $[G:H]\geq rank(G)$, one can always find a left-right transversal of $H$ which…

A classification of spanning surfaces for alternating links is provided up to genus, orientability, and a new invariant that we call aggregate slope. That is, given an alternating link, we determine all possible combinations of genus,…

Let $S$ be a compact orientable surface, and $\Mod(S)$ its mapping class group. Then there exists a constant $M(S)$, which depends on $S$, with the following property. Suppose $a,b \in \Mod(S)$ are independent (i.e., $[a^n,b^m]\not=1$ for…

In this paper, we study generating series enumerating polygonal angulations of closed oriented surfaces of fixed genus, focusing on $b$-angulations with $b = 3$ or $b = 2\nu$, $\nu \geq 2$. Based on Toda integrability, we establish new…

Given a graph embedded in an orientable surface, a process consisting of random excitations and random node and face balancing is constructed and analyzed. It is shown that given a priori bounds g' on the genus and n' on the number of…

Given a certain triangulation of a punctured surface with boundary, we construct a new triangulated surface without punctures which covers it. This new surface is naturally equipped with an action of a group of order two, and its quotient…

The bondage number $b(G)$ of a graph $G$ is the smallest number of edges whose removal from $G$ results in a graph with larger domination number. An orientable surface $\mathbb{S}_h$ of genus $h$, $h \geq 0$, is obtained from the sphere…

We restrict our discussion to the orientable category. For $g > 1$, let $OE_g$ be the maximum order of a finite group $G$ acting on the closed surface $\Sigma_g$ of genus $g$ which extends over $(S^3, \Sigma_g)$, where the maximum is taken…

This paper is about cohomology of mapping class groups from the perspective of arithmetic groups. For a closed surface $S$ of genus $g$, the mapping class group $Mod(S)$ admits a well-known arithmetic quotient $Mod(S)\rightarrow Sp(2g, Z)$,…

Let $C$ be a trigonal curve of genus $g\ge 5$ and let $T$ be the unique trigonal line bundle inducing a map $\pi: C \stackrel{3:1}{\longrightarrow} \mathbb{P}^1$. This note provides a short and easy proof of the normal generation for the…

Gross-Pandharipande-Siebert have shown that the 2-dimensional Kontsevich-Soibelman scattering diagrams compute certain genus zero log Gromov-Witten invariants of log Calabi-Yau surfaces. We show that the $q$-refined 2-dimensional…

A subset $S$ of a group $G$ invariably generates $G$ if $G= \langle s^{g(s)} | s \in S\rangle$ for every choice of $g(s) \in G,s \in S$. We say that a group $G$ is invariably generated if such $S$ exists, or equivalently if $S=G$ invariably…

Let (X, t, S) be a triple, where S is a compact, connected surface without boundary, and t is a free cellular involution on a CW-complex X. The triple (X, t, S) is said to satisfy the Borsuk-Ulam property if for every continuous map…

In this paper, we develop the theory of punctured R-maps as a crucial component of logarithmic gauged linear sigma models (log GLSM). A punctured R-map is a punctured map in the sense of ACGS, further twisted by the sheaf of differentials…

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…

Let $V$ be a plane smooth cubic curve over a finitely generated field $k.$ The Mordell-Weil theorem for $V$ states that there is a finite subset $P\subset V(k)$ such that the whole $V(k)$ can be obtained from $P$ by drawing secants and…