Compactifying the Space of Length Functions of a Right-angled Artin Group
Compactifying the Space of Length Functions of a Right-angled Artin Group
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直角Artin群长度函数空间的紧化
DOI：
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发表时间：
2015
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通讯作者：
A. Vijayan
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文献类型：
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作者：
A. Vijayan
卡勒和摩根证明了树上一个群的最小作用的长度函数完全决定了这个作用。因此，树上自由群的最小动作空间，直到比例(也称为外层空间)，通过将动作发送到其射影长度函数的映射嵌入到无限射影空间中。他们还证明了该映射的像具有紧闭包。
对于定义图是连通且无三角形的直角Artin群，我们研究了二维CAT(0)矩形复形上的极小作用空间。Charney和MarGolis证明了这种作用完全由它们的长度函数决定，因此这个空间嵌入到无限射影空间中。本文证明了嵌入映射的像，即与这些作用相关的射影长度函数集具有紧闭包。
Culler and Morgan proved that the length function of a minimal action of a group on a tree completely determines the action. As a consequence the space of minimal actions of a free group on trees, up to scaling (also known as Outer Space), embeds in infinite projective space via the map sending an action to its projectivized length function. They also proved that the image of this map has compact closure.
For a right-angled Artin group whose defining graph is connected and triangle-free, we investigate the space of minimal actions on 2-dimensional CAT(0) rectangle complexes. Charney and Margolis showed that such actions are completely determined by their length functions; hence this space embeds in infinite projective space. Here it is shown that the image of the embedding map, that is, the set of projectivized length functions associated to these actions, has compact closure.