Related papers: Generating mapping class group by two torsion elem…

Studying the virtual Euler characteristic of the moduli space of curves, Harer and Zagier compute the generating function $C_g(z)$ of unicellular maps of genus $g$. They furthermore identify coefficients, $\kappa^{\star}_{g}(n)$, which…

Let $F_g$ denote a closed oriented surface of genus $g$. A set of simple closed curves is called a filling of $F_g$ if its complement is a disjoint union of discs. The mapping class group $\text{Mod}(F_g)$ of genus $g$ acts on the set of…

We introduce machinery to allow ``cut-and-paste''-style inductive arguments in the Torelli subgroup of the mapping class group. In the past these arguments have been problematic because restricting the Torelli group to subsurfaces gives…

Let $M$ be a closed surface. By $\Homeo(M)$ we denote the group of orientation preserving homeomorphisms of $M$ and let $\MC(M)$ denote the Mapping class group. In this paper we complete the proof of the conjecture of Thurston that says…

Associated to any closed subgroup $G\subset U_N^+$ is a family of toral subgroups $T_Q\subset G$, indexed by the unitary matrices $Q\in U_N$. The family $\{T_Q|Q\in U_N\}$ is expected to encode the main properties of $G$, and there are…

Assume $k$ is a field and $R$ is a smooth $k$-algebra of dimension $d$. If $P$ is a projective module of rank $r$, then it is well-known that $P$ can be generated by $r+d$-elements (Forster--Swan). Under suitable assumptions on $r$ and $d$,…

We continue the study of low dimensional linear representations of mapping class groups of surfaces initiated by Franks--Handel and Korkmaz. We consider $(2g+1)$-dimensional complex linear representations of the pure mapping class groups of…

Finite presentations for the mapping class group M(F) are known for arbitrary orientable compact surface F. If F is non-orientable, then such presentations are known only when F has genus at most 3 and few boundary components. In this paper…

In this article we describe an algorithm that can be applied for the generation of various classes of maps on orientable surfaces. It uses existing generators for abstract graphs and combines them with an efficient embedding and isomorphism…

Suppose that $f$ is a homomorphism from the mapping class group $\mathcal{M}(N_{g,n})$ of a nonorientable surface of genus $g$ with $n$ boundary components, to $\mathrm{GL}(m,\mathbb{C})$. We prove that if $g\ge 5$, $n\le 1$ and $m\le g-2$,…

In [Tohoku Math. J. 62 (2010), 45--53] the second author showed that, except for a few cases, the order $N$ of a cyclic group of self-homeomorphisms of a closed orientable topological surface $S_g$ of genus $g \geq 2$ determines the group…

It is proved in [21] that there is a constant $c$ such that each transitive permutation group of degree $d\ge 2$ can be generated by $\lfloor cd/\sqrt{\log{d}}\rfloor$ elements. In this paper, we explicitly estimate $c$.

We consider the function $f(g)$ that assigns to an orientable surface $M$ of genus $g$ the maximal number of free commuting independent involutions on $M$. We show that the surface of minimal genus $g$ with $f(g)=n$ is a real moment-angle…

For a non-abelian group $G$, its commuting conjugacy class graph $\mathcal{CCC}(G)$ is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$…

Every finite simple group can be generated by two elements and, in fact, every nontrivial element is contained in a generating pair. Groups with this property are said to be $\frac{3}{2}$-generated, and the finite $\frac{3}{2}$-generated…

In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer representation of the braid group B_n,…

The Ptolemy groupoid is a combinatorial groupoid generated by elementary moves on marked trivalent fatgraphs with three types of relations. Through the fatgraph decomposition of Teichm\"uller space, the Ptolemy groupoid is a mapping class…

In this article we consider compact Riemann surfaces that are uniquely determined by the property of possessing a group of automorphisms of a prescribed order, strengthening uniqueness results proved by Nakagawa. More precisely, we deal…

We establish an explicit upper bound B(p,l,m), depending on p,l,m, on the number of conjugacy classes of order p^2 torsion elements u of type <l,m> of the Nottingham group defined over the prime field of characteristic p >0. In the cases…

We prove that the mapping class group of a closed oriented surface of genus at least two does not have Kazhdan's property (T).