论文标题
某些分段投影同构的有限属性
Finiteness properties of some groups of piecewise projective homeomorphisms
论文作者
论文摘要
Lodha-Moore Group $ g $首先出现，作为对冯·诺伊曼（Von Neumann）的猜想的有限反例。该组$ g $通过分段投影型自杀式以单位间隔作用。 lodha的结果表明，$ g $实际上具有$ f _ {\ infty} $。 在这里，我们将在作者与休斯的联合工作的意义上将$ g $描述为一个由反向半群$ s_ {2} $“本地确定”的组。 Semigroup $ s_ {2} $由三个线性分数转换$ a $，$ b $和$ c_ {2} $生成，其中$ a $和$ b $是双曲机平面的椭圆形变换，$ c_ {2} $是一种超轻的翻译。按照Farley和Hughes划定的一般过程，我们提供了一个新的证明，即$ G $具有$ f _ {\ infty} $。我们的证明同时表明，作用在行，圆圈和cantor集的各个组具有$ f _ {\ infty} $。我们还证明了由反向半群$ s_ {3} $在本地确定的群体的类似结果，该组与$ s_ {2} $共享发电机$ a $ a $ a $ a $ a $ a $ a $ a $ a {2} $，但用不同的高压转换$ c_ {3} $代替$ c_ {2} $。
The Lodha-Moore group $G$ first arose as a finitely presented counterexample to von Neumann's conjecture. The group $G$ acts on the unit interval via piecewise projective homemorphisms. A result of Lodha shows that $G$ in fact has type $F_{\infty}$. Here we will describe $G$ as a group that is "locally determined" by an inverse semigroup $S_{2}$, in the sense of the author's joint work with Hughes. The semigroup $S_{2}$ is generated by three linear fractional transformations $A$, $B$, and $C_{2}$, where $A$ and $B$ are elliptical transformations of the hyperbolic plane and $C_{2}$ is a hyperbolic translation. Following a general procedure delineated by Farley and Hughes, we offer a new proof that $G$ has type $F_{\infty}$. Our proof simultaneously shows that various groups acting on the line, the circle, and the Cantor set have type $F_{\infty}$. We also prove analogous results for the groups that are locally determined by an inverse semigroup $S_{3}$, which shares the generators $A$ and $B$ with $S_{2}$, but replaces $C_{2}$ with a different hyperbolic translation $C_{3}$.
