Lodha-Moore群的BNSR-不变量和$\textrm{F}_\infty$型奇异单群
The BNSR-invariants of the Lodha-Moore groups, and an exotic simple group of type $\textrm{F}_\infty$
摘要：
本文给出了Lodha-Moore群的Bieri-Neumann-Strebel-Renz不变量的完整刻画。第二作者之前计算了前两个不变量，在这里我们证明了所有更高的不变量与第二个一致，完成了完整的计算。作为结果，我们给出了第一Lodha-Moore群的正规子群的有限性的完整图象.特别地，我们证明了该群的每一个正规子群都是$\textrm{F}_\infty$型的，回答了Oberwolfach Rep.的问题112，15（2）：1579 - 1633，2018。证明涉及应用Bestvina-Brady离散Morse理论的变化，所谓的集群复杂$X$介绍的第一作者。作为应用，我们还证明了第一作者以前构造的某个单群$S$是$\textrm{F}_\infty$型的.这提供了$\textrm{F}_\infty$型单群的第一个例子，它通过同胚忠实地作用在圆上，但不允许通过$C^1$-同胚或通过分段线性同胚在任何$1$-流形上的任何非平凡作用。
英文摘要：
In this paper we give a complete description of the Bieri-Neumann-Strebel-Renz invariants of the Lodha-Moore groups. The second author previously computed the first two invariants, and here we show that all the higher invariants coincide with the second one, which finishes the complete computation. As a consequence, we present a complete picture of the finiteness properties of normal subgroups of the first Lodha-Moore group. In particular, we show that every finitely presented normal subgroup of the group is of type $\textrm{F}_\infty$, answering question 112 from Oberwolfach Rep., 15(2):1579-1633, 2018. The proof involves applying a variation of Bestvina-Brady discrete Morse theory to the so called cluster complex $X$ introduced by the first author. As an application, we also demonstrate that a certain simple group $S$ previously constructed by the first author is of type $\textrm{F}_\infty$. This provides the first example of a type $\textrm{F}_\infty$ simple group that acts faithfully on the circle by homeomorphisms, but does not admit any nontrivial action by $C^1$-diffeomorphisms, nor by piecewise linear homeomorphisms, on any $1$-manifold.
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