Related papers: Every mapping class group is generated by 6 involu…

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. In this paper, we derive necessary and sufficient conditions under which two torsion elements in $\mathrm{Mod}(S_g)$ will have…

We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends space, as defined by Mann and Rafi, and…

The purpose of this paper is to prove that if $G$ is a transitive permutation group of degree $n\geq 2$, then $G$ can be generated by $\lfloor cn/\sqrt{\log{n}}\rfloor$ elements, where $c:=\sqrt{3}/2$. Owing to the transitive group…

A (2,*)-group is a group that can be generated by two elements, one of which is an involution. We describe the method we have used to produce a census of all (2,*)-groups of order at most 6 000. Various well-known combinatorial structures…

Let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. In this paper, we derive necessary and sufficient conditions for two finite-order mapping classes to have commuting conjugates in…

A topological group $G$ is topologically normally generated if there exists $g \in G$ such that the normal closure of $g$ is dense in $G$. Let $S$ be a tame, infinite type surface whose mapping class group $\mathrm{Map}(S)$ is generated by…

We study the quotient of the mapping class group $\operatorname{Mod}_g^n$ of a surface of genus $g$ with $n$ punctures, by the subgroup $\operatorname{Mod}_g^n[p]$ generated by the $p$-th powers of Dehn twists. Our first main result is that…

Faltings' theorem [Fal83],[Fal91] (formerly the Mordell conjecture [Mo22]) states that a curve of genus greater than one over any number field has only finitely many points. Again a natural question is how many points can such a curve have.…

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g \geq 1$, and let $\mathrm{LMod}_{p}(X)$ be the liftable mapping class group associated with a finite-sheeted branched cover $p:S \to X$, where…

For $g\geq 2$, let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g$. In this paper, we provide necessary and sufficient conditions for the existence of infinite metacyclic subgroups of…

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 obtain a minimal generating set of involutions for the level 2 subgroup of the mapping class group of a closed nonorientable surface.

Wajnryb proved that the mapping class group of a closed oriented surface is generated by two elements. We proved that the mapping class group is generated by two pseudo-Anosov elements. In particular, if the genus is greater than or equal…

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…

In this paper, we prove that each automorphism of the Torelli group of a surface is induced by a diffeomorphism of the surface, provided that the surface is a closed, connected, orientable surface of genus at least 3. This result was…

Let $S_g$ be a closed, oriented surface of genus $g$, and let $\operatorname{Mod}(S_g)$ denote its mapping class group. The Torelli group $\mathcal{I}_g$ is the subgroup of $\operatorname{Mod}(S_g)$ consisting of mapping classes that act…

A subset $\left\{x_{1},x_{2},\hdots,x_{d}\right\}$ of a group $G$ \emph{invariably generates} $G$ if $\left\{x_{1}^{g_{1}},x_{2}^{g_{2}},\hdots,x_{d}^{g_{d}}\right\}$ generates $G$ for every $d$-tuple $(g_{1},g_{2}\hdots,g_{d})\in G^{d}$.…

In this paper we prove a theorem stated by Castelnuovo which bounds the dimension of linear systems of plane curves in terms of two invariants, one of which is the genus of the curves in the system. Then we classify linear systems whose…

We provide a simple criterion for an element of the mapping class group of a closed surface to have normal closure equal to the whole mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a…

Let $S_g$ be the closed oriented surface of genus $g \geq 0$, and let $\mathrm{Mod}(S_g)$ be the mapping class group of $S_g$. For $g\geq 2$, we develop an algorithm to obtain a finite generating set for the liftable mapping class group…