Related papers: Automorphisms of braid groups on orientable surfac…

We have proved that the group of all inner automorphisms of the free Burnside group $B(m,n)$ is the unique normal subgroup in $Aut(B(m,n))$ among all its subgroups, which are isomorphic to free Burnside group $B(s,n)$ of some rank $s$ for…

We study automorphism and birational automorphism groups of varieties over fields of positive characteristic from the point of view of Jordan and $p$-Jordan property. In particular, we show that the Cremona group of rank $2$ over a field of…

Motivated by the recently introduced concept of a pseudosymmetric braided monoidal category, we define the pseudosymmetric group PS_n, as the quotient of the braid group B_n by the relations \sigma_i\sigma_{i+1}^{-1}\sigma_i=\sigma…

Let $\Gamma$ be a finite connected graph. The (unlabelled) configuration space $UC^n \Gamma$ of $n$ points on $\Gamma$ is the space of $n$-element subsets of $\Gamma$. The $n$-strand braid group of $\Gamma$, denoted $B_n\Gamma$, is the…

Let $\Sigma$ be a bounded surface. We prove the Dehn-Nielsen-Baer theorem for bounded surfaces to show that the mapping class group of $\Sigma$ is isomorphic to the automorphisms of the fundamental groupoid of $\Sigma$ that fix loops around…

We study a novel type of braid groups on a closed orientable surface $\Sigma$. These are fundamental groups of certain manifolds that are hybrids between symmetric products and configuration spaces of points on $\Sigma$; a class of examples…

We show that branched coverings of surfaces of large enough genus arise as characteristic maps of braided surfaces that is, lift to embeddings in the product of the surface with $\mathbb R^2$. This result is nontrivial already for…

Let $G$ be a group. The orbits of the natural action of $Aut(G)$ on $G$ are called the automorphism orbits of $G$, and their number is denoted by $\omega(G)$. Let $\mathbb{F}$ be an infinite field, and let $UT_n(\mathbb{F})$ denote the…

A closed Riemann surface $S$ (of genus at least one) is called an origami curve if it admits a non-constant holomorphic map $\beta:S \to E$ with at most one branch value, where $E$ is a genus one Riemann surface. In this case, $(S,\beta)$…

Let $\Gamma_{g,1}^m$ be the mapping class group of the orientable surface $\Sigma_{g,1}^m$ of genus $g$ with one parametrised boundary curve and $m$ permutable punctures; when $m=0$ we omit it from the notation. Let…

We consider translation surfaces with poles on surfaces. We shall prove that any finite group appears as the automorphism group of some translation surface with poles. As a direct consequence we obtain the existence of structures achieving…

Let $X$ be a smooth hypersurface of dimension $n\geq 1$ and degree $d\geq 3$ in the projective space given as the zero set of a homogeneous form $F$. If $(n,d)\neq (1,3), (2,4)$ it is well known that every automorphism of $X$ extends to an…

From a group $H$ and a non-trivial element $h$ of $H$, we define a representation $\rho: B_n \to \Aut(G)$, where $B_n$ denotes the braid group on $n$ strands, and $G$ denotes the free product of $n$ copies of $H$. Such a representation…

Let $F(x, y)$ be a binary form of degree at least three and non-zero discriminant. In this article we compute the automorphism group $\operatorname{Aut} F$ for four families of binary forms. The first two families that we are interested in…

The braided Thompson group $\mathcal B$ is an asymptotic mapping class group of a sphere punctured along the standard Cantor set, endowed with a rigid structure. Inspired from the case of finite type surfaces we consider a Hatcher-Thurston…

Herein we prove that if $M$ is a compact oriented Riemann surface of genus $g$, and $M^{[n]}$ is the classifying space of $n$ distinct, unordered points on $M$, then the kernel of the map $\pi_1(M^{[n]})\to H_1(M)$ is generated by…

A metacyclic group $H$ can be presented as $\langle \alpha,\beta\mid \alpha^{n}=1, \ \beta^{m}=\alpha^{t}, \ \beta\alpha\beta^{-1}=\alpha^{r}\rangle$ for some $n,m,t,r$. Each endomorphism $\sigma$ of $H$ is determined by…

A countable graph is ultrahomogeneous if every isomorphism between finite induced subgraphs can be extended to an automorphism. Woodrow and Lachlan showed that there are essentially four types of such countably infinite graphs: the random…

Let $D$ be a simple derivation of the polynomial ring $\mathbb{k}[x_1,\dots,x_n]$, where $\mathbb{k}$ is an algebraically closed field of characteristic zero, and denote by…

This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic…