论文标题
在$ k $ - 地理图和组上
On $k$-geodetic graphs and groups
论文作者
论文摘要
我们称之为图$ k $ - 地理器，如果连接了一些$ k \ geq 1 $，并且在任何两个顶点之间最多都有$ k $ Geodesics。结果表明，任何具有$ k $ - 地理cayley图的双曲线组几乎没有。此外，在这样的组中，任何无限级元素的中心者都是无限的循环基团。这些结果以前仅在$ k = 1 $的情况下才知道。用于开发该定理的关键工具是关于$ K $地理图中``梯子般的结构'''的新图理论结果。
We call a graph $k$-geodetic, for some $k\geq 1$, if it is connected and between any two vertices there are at most $k$ geodesics. It is shown that any hyperbolic group with a $k$-geodetic Cayley graph is virtually-free. Furthermore, in such a group the centraliser of any infinite order element is an infinite cyclic group. These results were known previously only in the case that $k=1$. A key tool used to develop the theorem is a new graph theoretic result concerning ``ladder-like structures'' in a $k$-geodetic graph.
