Asymptotically rigid mapping class groups, I : Finiteness properties of braided Thompson’s and Houghton’s groups
Asymptotically rigid mapping class
groups, I : Finiteness properties of braided Thompson’s and Houghton’s
groups
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非对称刚性映射类
DOI：
10.2140/gt.2022.26.1385
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
Christian Urech
中科院分区：
文献类型：
--
作者：
A. Genevois;Anne Lonjou;Christian Urech
本文致力于研究通过加厚平面树得到的无穷穿孔曲面的渐近刚性映射类群。这些群包括由Funar和Kapoudkin引入的编织的Ttolemy-Thompson群，以及由Degenhardt引入的编织的Houghton群。我们给出了一个可压缩立方体复形的初等构造，这些群作用于它上的立方体稳定器同构于辫子群的有限扩张。作为应用，我们证明了Funar-Kapoudkin和Degenhardt的猜想，证明了$T^，T^ast$是$F_inty$型，而$mathrm{br}H_n$是$F_(n-1)$型，但不是$F_n$型。
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 $T^\sharp,T^\ast$ are of type $F_\infty$ and that $\mathrm{br}H_n$ is of type $F_{n-1}$ but not of type $F_n$.