论文标题
在有限生成的基团中的阳性锥的几何形状上
On the geometry of positive cones in finitely generated groups
论文作者
论文摘要
我们研究了有限生成的组中剩余的总订单（左顺序）的正锥的几何形状。我们介绍了\ textit {hucha属性}和\ dexit {prieto property}，以供左订购组。第一个意味着在任何左顺阶中，相应的阳性锥体都不是粗体连接的，而在任何左顺序中，相应的阳性圆锥体的第二个都可以切好。我们表明，所有可左订购的免费产品均具有Hucha属性，并且Hucha属性在某些免费产品中具有与Prieto子组合并的某些免费产品稳定。作为一个应用程序，我们表明，从Z. Sela（例如自由组，基本双曲线表面，自由小组的双打等基本组）和非亚伯利亚有限生成的免费$ \ mathbb {Q} $组的非亚伯利亚子组的非亚伯利亚子组的非亚伯利亚的亚组 - Baumsslag the hucha property中。特别是，这意味着这些组具有空的BNS-Invariant $σ^1 $，并且它们没有有限产生的正锥。
We study the geometry of positive cones of left-invariant total orders (left-order, for short) in finitely generated groups. We introduce the \textit{Hucha property} and the \texit{Prieto property} for left-orderable groups. The first one means that in any left-order the corresponding positive cone is not coarsely connected, and the second one that in any left-order the corresponding positive cone is coarsely connected. We show that all left-orderable free products have the Hucha property, and that the Hucha property is stable under certain free products with amalgamatation over Prieto subgroups. As an application we show that non-abelian limit groups in the sense of Z. Sela (e.g. free groups, fundamental group of hyperbolic surfaces, doubles of free groups and others) and non-abelian finitely generated subgroups of free $\mathbb{Q}$-groups in the sense of G. Baumslag have the Hucha property. In particular, this implies that these groups have empty BNS-invariant $Σ^1$ and that they don't have finitely generated positive cones.
