Related papers: Isometric point-circle configurations on surfaces …

We prove that every endomorphism of the mapping class group of an orientable surface onto a subgroup of finite index is in fact an automorphism.

We investigate the automorphism group of the substructure ordering of finite directed graphs. The second author conjectured that it is isomorphic to the 768-element group $(\mathbb{Z}_2^4 \times S_4)\rtimes_{\alpha} \mathbb{Z}_2$. Though…

We show that if $\cal S$ is a compact Riemann surface of genus $g = p+1$, where $p$ is prime, with a group of automorphisms $G$ such that $|G|\geq\lambda(g-1)$ for some real number $\lambda>6$, then for all sufficiently large $p$ (depending…

We find normal forms for del Pezzo surfaces of degree $2$ over algebraically closed fields of characteristic $2$. For each normal form, we describe the structure of the group of automorphisms of the surface. In particular, we classify all…

Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the…

We classify the singular loci of real surfaces in three-space that contain two circles through each point. We characterize how a circle in such a surface meets this loci as it moves in its pencil and as such provide insight into the…

In this article we introduce new affinely invariant points---`special parabolic points'---on the parabolic set of a generic surface $M$ in real 4-space, associated with symmetries in the 2-parameter family of reflexions of $M$ in points of…

We show that there is a collection of subgroups of the mapping class group of a surface such that the associated coset intersection complex is quasi-isometric and homotopy equivalent to the curve complex. Moreover, we prove that these two…

We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of…

In this paper, we briefly review some of the known results concerning the cohomological structures of the mapping class group of surfaces, the outer automorphism group of free groups, the diffeomorphism group of surfaces as well as various…

{\it Fold maps} are fundamental tools in generalizing the theory of Morse functions and its application to studies of geometric properties of manifolds. One of the fundamental and important problems in the theory of fold maps is to…

For $a,b\geq 3$, we calculate the orders of automorphisms of smooth curves with bidegree $(a,b)$ in the product $\pp$ of the projective line $\mathbb P^1$. We identify smooth curves in $\pp$ which have automorphisms with the largest orders.…

We prove that the envelope of meromorphy of any imbedded symplectic sphere in $CP^2$ coincides with the whole $CP^2$. As a tool for the proof we use the Gromov theory of pseudo-holomorphic curves. Several results in this subject, such as…

We give a combinatorial description of closed curves on oriented surfaces in terms of certain permutations, called charts. We describe automorphisms of curves in terms of charts and compute the total number of curves counted with…

Let $p \geqslant 5$ be a prime number. In this article we provide a complete and explicit description of the locus formed by the compact Riemann surfaces of genus $2(p-1)$ that are endowed with a group of automorphisms of order $4p$. In…

If $p$ is prime, a compact Riemann surface $X$ of genus $g\geq 2$ is called cyclic $p$-gonal if it admits a cyclic group of automorphisms $C_{p}$ of order $p$ such that the quotient space $X/C_{p}$ has genus 0. If in addition $C_{p}$ is not…

In this paper, we show that the infinitesimal Torelli theorem implies the existence of deformations of automorphisms. In the first part, we use Hodge theory and deformation theory to study the deformations of automorphisms of complex…

We construct orbits of the absolute Galois group, of explicit unbounded size, consisting of surfaces with mutually non-isomorphic fundamental groups. These are Beauville surfaces with Beauville group PGL_2(p).

We study infinitesimal deformations of complete hyperbolic surfaces with boundary and with ideal vertices, possibly decorated with horoballs. ``Admissible'' deformations are the ones that pull all horoballs apart; they form a convex cone of…

Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of…