Limit groups, positive-genus towers and measure equivalence

From symmetries of the modular tower of genus zero real stable curves to an Euler class for the dyadic circle

We build actions of Thompson group V (related to the Cantor set) and of the so-called "spheromorphism" group of Neretin, on "towers" of moduli spaces of genus zero real stable curves. The latter consist of inductive limits of spaces which are the real parts of the Grothendieck-Knudsen compactification of the usual moduli spaces of punctured Riemann spheres. By a result of M. Davis, T. Januszkiewicz and R. Scott, these spaces are aspherical cubical complexes, whose fundamental groups, the "pure quasi-braid groups", are some analogues of the classical pure braid groups. By lifting the actions of…

关于映射类群二元生成元的 Nielsen 等价类

On Nielsen equivalence classes of two-elements generators of mapping class groups

本文研究闭有向曲面映射类群的二元生成元分类，证明亏格不小于八时存在无穷多个 Nielsen 等价类。

Towers and the first-order theory of hyperbolic groups

This paper is devoted to the first-order theory of torsion-free hyperbolic groups. One of its purposes is to review some results and to provide precise and correct statements and definitions, as well as some proofs and new results. A key concept is that of a tower (Sela) or NTQ system (Kharlampovich-Myasnikov). We discuss them thoroughly. We state and prove a new general theorem which unifies several results in the literature: elementarily equivalent torsion-free hyperbolic groups have isomorphic cores (Sela); if $H$ is elementarily embedded in a torsion-free hyperbolic group $G$, then $G$ is …

Coarse non-amenability and covers with small eigenvalues

Given a closed Riemannian manifold M and a (virtual) epimorphism from the fundamental group of M onto a free group of rank 2, we construct a tower of finite sheeted regular covers {M_n}_{n=0}^{\infty} of M such that the first non-zero eigenvalues λ_1(M_n) of the Laplacian converge to zero as n tends to infinity. This is the first example of such a tower which is not obtainable up to uniform quasi-isometry (or even up to uniform coarse equivalence) by the previously known methods where the fundamental group of M is supposed to surject onto an amenable group.

可定向曲面上有限群作用的等价类计算：动态综述

Computing equivalence classes of finite group actions on orientable surfaces: A dynamic survey

本文研究黎曼曲面上有限群作用的拓扑等价分类问题，通过富克斯群自同构作用分析并借助计算机代数系统，给出了亏格 g≤9 的完整分类列表。

柄体群的测度等价刚性

Measure equivalence rigidity of the handlebody groups

本文证明了亏格至少为3的3维连通柄体群在测度等价下具有超刚性，并推导出格嵌入、轨道等价及弱紧群作用的W*-刚性定理。