论文标题
Lodha-Moore组的BNSR不变，以及一个异国情调的简单类型$ \ textrm {f} _ \ infty $
The BNSR-invariants of the Lodha-Moore groups, and an exotic simple group of type $\textrm{F}_\infty$
论文作者
论文摘要
在本文中，我们对Lodha-Moore群体的Bieri-Neumann-Strebel-Renz不变性进行了完整的描述。第二位作者先前计算了前两个不变性，在这里我们表明，所有较高的不变性都与第二个不变式相吻合，这完成了完整的计算。结果，我们介绍了第一个Lodha-Moore组的正常亚组的有限属性的完整图片。特别是，我们表明，该组的每个有限的正常子组都是$ \ textrm {f} _ \ infty $，回答了Oberwolfach Rep。的问题112，15（2）：1579-1633，2018。证明涉及将Bestvina-Brady Morse Morse Themord的变化应用于$ x $ x $ x $ x $ x $ x，作为一个应用程序，我们还证明了第一作者先前构建的某个简单组$ s $是类型$ \ textrm {f} _ \ infty $。这提供了类型的$ \ textrm {f} _ \ infty $简单组的第一个示例，该组通过同构忠实地在圆圈上行事，但不承认$ c^1 $ -Diffeomorphormiss的任何非平凡的动作，也不是通过分段线性同音同态的$ 1 $ -1美元。
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.
