论文标题
在Cayley自动组的几何形状上
On the geometry of Cayley automatic groups
论文作者
论文摘要
与自动的相反，成为Cayley Automatic \ Emph {先验}没有几何后果。具体来说，自动团体的Cayley图表享受旅行者的同事。在这里，我们研究了第一作者和Trakuldit引入的距离功能，该功能旨在衡量Cayley Automatic Group距离自动的距离，因为Cayley Graph的旅行者属性失败了。第一作者和Trakuldit表明，如果它最多失败，那么该组实际上是自动的。在本文中，我们表明，对于一大批非自动cayley自动组，此函数以下面的线性函数在本文定义的精确含义中受到线性函数的限制。实际上，对于所有具有超季度DEHN函数或不有限呈现的Cayley自动组，我们可以构建一个非降低函数，该函数仅取决于组，并且（2）从距离函数的范围以及组上任何Cayley自动结构的距离。
In contrast to being automatic, being Cayley automatic \emph{a priori} has no geometric consequences. Specifically, Cayley graphs of automatic groups enjoy a fellow traveler property. Here we study a distance function introduced by the first author and Trakuldit which aims to measure how far a Cayley automatic group is from being automatic, in terms of how badly the Cayley graph fails the fellow traveler property. The first author and Trakuldit showed that if it fails by at most a constant amount, then the group is in fact automatic. In this article we show that for a large class of non-automatic Cayley automatic groups this function is bounded below by a linear function in a precise sense defined herein. In fact, for all Cayley automatic groups which have super-quadratic Dehn function, or which are not finitely presented, we can construct a non-decreasing function which (1) depends only on the group and (2) bounds from below the distance function for any Cayley automatic structure on the group.
