论文标题
对称空间和欧几里得建筑物的粗嵌入
Coarse embeddings of symmetric spaces and Euclidean buildings
论文作者
论文摘要
Gromov在80年代引入的粗糙嵌入是当控制函数不一定是仿射时的准等级嵌入的概括。在本文中，我们将对对称空间和欧几里得建筑物之间的粗嵌入特别感兴趣。由于Anderson-Schroeder，Kleiner，Kleiner-Leeb，Eskin-Farb和Fisher-Whyte的对称空间和更高等级的对称空间和更高等级的建筑物的僵化结果，因此对准时的情况非常了解。特别是，众所周知，这些空间的等级在准等级嵌入下是单调的。如霍斯层嵌入所示，粗嵌入不再是这种情况。但是，我们表明，在域中没有欧几里得因子的情况下，在粗嵌入下，等级是单调的。这回答了戴维·费舍尔（David Fisher）和凯文·怀特（Kevin Whyte）的问题。当我们用适当的CoCocact Cat（0）空间或映射类组替换目标空间时，这仍然存在。在对称空间和欧几里得建筑物之间，我们还可以通过允许其含有尺寸1的欧几里得因素1来放松域上的状况，从而回答了Gromov的问题。
Introduced by Gromov in the 80's, coarse embeddings are a generalization of quasi-isometric embeddings when the control functions are not necessarily affine. In this paper, we will be particularly interested in coarse embeddings between symmetric spaces and Euclidean buildings. The quasi-isometric case is very well understood thanks to the rigidity results for symmetric spaces and buildings of higher rank by Anderson-Schroeder, Kleiner, Kleiner-Leeb, Eskin-Farb and Fisher-Whyte. In particular, it is well known that the rank of these spaces is monotonous under quasi-isometric embeddings. This is no longer the case for coarse embeddings as shown by horospherical embeddings. However, we show that in the absence of a Euclidean factor in the domain, the rank is monotonous under coarse embeddings. This answers a question by David Fisher and Kevin Whyte. This still holds when we replace the target space by a proper cocompact CAT(0) space or by a mapping class group. Between symmetric spaces and Euclidean buildings, we can also relax the condition on the domain by allowing it to contain a Euclidean factor of dimension 1, answering a question by Gromov.
