Automorphisms of the Complex of Curves

Eigenforms of hyperelliptic curves with many automorphisms

Given a pair of translation surfaces it is very difficult to determine whether they are supported on the same algebraic curve. In fact, there are very few examples of such pairs. In this note we present infinitely many examples of finite collections of translation surfaces supported on the same algebraic curve. The underlying curves are hyperelliptic curves with many automorphisms. For each curve, the automorphism of maximal order acts on the space of holomorphic 1-forms. We present a translation surface corresponding to each of the eigenforms of this action.

Automorphism groups of origami curves

A closed Riemann surface $S$ (of genus at least one) is called an origami curve if it admits a non-constant holomorphic map $β:S \to E$ with at most one branch value, where $E$ is a genus one Riemann surface. In this case, $(S,β)$ is called an origami pair and ${\rm Aut}(S,β)$ is the group of conformal automorphisms $ϕ$ of $S$ such that $β=β\circ ϕ$. Let $G$ be a finite group. It is a known fact that $G$ can be realized as a subgroup of ${\rm Aut}(S,β)$ for a suitable origami pair $(S,β)$. It is also known that $G$ can be realized as a group of conformal automorphisms of a Riemann surface $X$ …

Automorphisms of the Torelli complex and the complex of separating curves

We compute the automorphism groups of the Torelli complex and the complex of separating curves for all but finitely many compact orientable surfaces. As an application, we show that the abstract commensurators of the Torelli group and the Johnson kernel for such surfaces are naturally isomorphic to the extended mapping class group.

Strongly irreducible surface automorphisms

A surface automorphism is strongly irreducible if every essential simple closed curve in the surface has nontrivial geometric intersection with its image. We show that a three-manifold admits only finitely many inequivalent surface bundle structures with strongly irreducible monodromy.

Automorphism groups of simplicial complexes of infinite type surfaces

Let S be any orientable surface of infinite genus with a finite number of boundary components. In this work we consider the curve complex C(S), the nonseparating curve complex N(S) and the Schmutz graph G(S) of S. When all the topological ends of S carry genus, we show that all elements in the automorphism groups Aut(C(S)), Aut(N(S)) and Aut(G(S)) are geometric, i.e. these groups are naturally isomorphic to the extended mapping class group MCG*(S) of the infinite surface S. Finally, we study rigidity phenomena within Aut(C(S)) and Aut(N(S))

procongruence曲线复形与裤子复形的自同构

Automorphisms of procongruence curve and pants complexes

本文通过研究procongruence映射类群在曲线和裤子复形上的作用，证明了裤子复形procongruence完备化的刚性定理，并由此得出光滑代数曲线模栈满足弱anabelan性质。

k-曲线图的自同构

Automorphisms of the k-curve graph

本文研究有限型可定向曲面上的 k-曲线图，证明当 k 相对欧拉示性数足够小时其自同构群同构于扩展映射类群，并将结论推广至 systolic complex，解决 Schmutz Schaller 的猜想。

通过锥猜想建立自同构群的几何有限性

Geometrical finiteness for automorphism groups via cone conjecture

本文通过锥猜想研究K3、恩里ques、Coble表面及不可约辛流形的自同构群在超球面上的等距作用的几何有限性，证明此类群非正曲率特性，并揭示K3表面的Kleinian晶格与(-2)曲线关系。