Gregory C. Bell, Alexander Dranishnikov
By recognizing them as fundamental groups of developable complexes of groups we prove that mapping class groups of compact orientable surfaces have finite asymptotic dimension.
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An explicit presentation for asymptotically rigid mapping class groups
本文通过图群的fundamental群商，显式构造了若干渐近刚性映射类群，并基于Labruère和Paris的工作，计算了表面渐近刚性映射类群的显式表示。
Even though big mapping class groups are not countably generated, certain big mapping class groups can be generated by a coarsely bounded set and have a well defined quasi-isometry type. We show that the big mapping class group of a stable surface of infinite type with a coarsely bounded generating set that contains an essential shift has infinite asymptotic dimension. This is in contrast with the mapping class groups of surfaces of finite type where the asymptotic dimension is always finite. We also give a topological characterization of essential shifts.
The main goal of this paper is a detailed study of asymptotic cones of the mapping class groups. In particular, we prove that every asymptotic cone of a mapping class group has a bi-Lipschitz equivariant embedding into a product of real trees, sending limits of hierarchy paths onto geodesics, and with image a median subspace. One of the applications is that a group with Kazhdan's property (T) can have only finitely many pairwise non-conjugate homomorphisms into a mapping class group. We also give a new proof of the rank conjecture of Brock and Farb (previously proved by Behrstock and Minsky, a…
Asymptotic mapping class groups of Cantor manifolds and their finiteness properties
该研究证明了特定曲面渐近映射类群的$F_\infty$型有限性，提出更一般的Cantor流形渐近映射类群$F_\infty$型判定定理，给出多类重要示例并验证了部分流形下其同调与稳定同调的一致性。
CAT(0) cube complexes and asymptotically rigid mapping class groups
本文研究由平面树加厚得到的无限穿孔曲面的渐近刚性映射类群，构造了一族其上可作用的 CAT(0) 立方复形，并确定了 Genevois、Lonjou 和 Urech 早先引入的立方复形为 CAT(0) 的条件。
Asymptotically rigid mapping class groups II: strand diagrams and nonpositive curvature
该研究提出由特定半群表示的编织链图构造的Chambord群新类，涵盖多种已知渐近刚性映射类群，并证明其多循环子群具有几乎阿贝尔且不失真的性质。
In this work, we study the asymptotic geometry of the mapping class group and Teichmueller space. We introduce tools for analyzing the geometry of `projection' maps from these spaces to curve complexes of subsurfaces; from this we obtain information concerning the topology of their asymptotic cones. We deduce several applications of this analysis. One of which is that the asymptotic cone of the mapping class group of any surface is tree-graded in the sense of Drutu and Sapir; this tree-grading has several consequences including answering a question of Drutu and Sapir concerning relatively hype…
We consider Thompson's groups from the perspective of mapping class groups of surfaces of infinite type. This point of view leads us to the braided Thompson groups, which are extensions of Thompson's groups by infinite (spherical) braid groups. We will outline the main features of these groups and some applications to the quantization of Teichmüller spaces. The chapter provides an introduction to the subject with an emphasis on some of the authors results.
This article is dedicated to the study of asymptotically rigid mapping class groups of infinitely-punctured surfaces obtained by thickening planar trees. Such groups include the braided Ptolemy-Thompson groups $T^\sharp,T^\ast$ introduced by Funar and Kapoudjian, and the braided Houghton groups $\mathrm{br}H_n$ introduced by Degenhardt. We present an elementary construction of a contractible cube complex, on which these groups act with cube-stabilisers isomorphic to finite extensions of braid groups. As an application, we prove Funar-Kapoudjian's and Degenhardt's conjectures by showing that $T…
Asymptotically rigid mapping class groups of infinite graphs
本文引入无限图的渐近刚性映射类群，刻画其有限性性质与图端数的关系，给出纯图 Houghton 群的显式展示，并证明其构成区别于既有 Houghton 型群的新群类。
Asymptotically rigid mapping class groups III: Presentations and isomorphisms
本文通过群在脊柱复形上的作用，计算了编织Higman-Thompson群的明确展示及其阿贝尔化，并利用这些代数不变量解决了树的渐近刚性映射类群同构问题的多个新情况。
Asymptotically conformal and asymptotically rigid mapping class groups
本文给出渐近刚性映射类群在渐近共形模群中有限指数的条件，并由此证明当曲面具有n≥3个端时模群具有类型F_{n-1}但不具有FP_n，同时推广到L^p模群。
We show that a random walk on the mapping class group of an orientable surface gives rise to a pseudo-Anosov element with asymptotic probability one. Our methods apply to many subgroups of the mapping class group, including the Torelli group.
Geometric models and asymptotic dimension for infinite-type surface mapping class groups
本文为无限型曲面映射类群构造了简单弧线与曲线的度量图，证明了其粗等价与拟等距性质，并完成了该类群渐近维数的分类。
Finiteness properties of asymptotically rigid handlebody groups
本文引入渐近刚性柄体映射类群，研究其有限性性质如何依赖于柄体的端空间，并证明在某些情形下其同调与柄体群的稳定同调一致。
A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary (relatively) hyperbolic groups, rank 1 CAT(0) groups, mapping class groups and Out(Fn). As an application, we show that mapping class groups act on finite products of δ-hyperbolic spaces so that orbit maps are quasi-isometric embeddings. We prove that mapping class groups have finite asymptotic dimension.
Normal generators for mapping class groups
研究测地群的正则生成元与Teichmüller空间和曲率图上渐近翻译长度的关系，探讨相关开放问题。
We consider the hyperelliptic handlebody group on a closed surface of genus $g$. This is the subgroup of the mapping class group on a closed surface of genus $g$ consisting of isotopy classes of homeomorphisms on the surface that commute with some fixed hyperelliptic involution and that extend to homeomorphisms on the handlebody. We prove that the logarithm of the minimal dilatation (i.e, the minimal entropy) of all pseudo-Anosov elements in the hyperelliptic handlebody group of genus $g$ is comparable to $1/g$. This means that the asymptotic behavior of the minimal pseudo-Anosov dilatation of…
Reducible normal generators for mapping class groups are abundant
该研究探讨映射类群的正规生成问题，给出映射类成为正规生成元的条件，提出基于Teichmüller空间渐近平移长度的可约映射类正规生成判别准则，类比了Lanier-Margalit的伪Anosov生成元相关研究。
An asymptotic for the Hall--Paige conjecture
本文用解析数论圆法给出满足Hall–Paige条件的n阶群G的完全映射数目的渐近公式(e^{-1/2}+o(1))|G^ab|n!^2/n^n，为Hall–Paige猜想提供了新的量化证明途径。
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