论文标题
模型有限生成的组的几何形状
Model geometries of finitely generated groups
论文作者
论文摘要
我们研究有限生成的组的几何形状。如果有限生成的组不包含非平凡的有限等级的无用Abelian相称的子组，我们显示任何模型几何形状都由非连接类型的对称空间，无限的本地有限的顶点传播图形或此类空间的产物主导。我们还证明，有限生成的组具有不受局部图的模型几何形状，并且仅当它包含一个相称的有限等级的无悬浮式Abelian子组或均匀的相称的亚组，该子组是半imimple Lie组中均匀的晶格。这表征了有限生成的基团，这些组嵌入了均匀的晶格中，而不是逐个紧凑的组（完全断开连接）。我们表明，唯一的共同体两组是表面组和广义的鲍姆斯拉格 - 统计组，我们获得了同一个尺寸三组的类似表征。
We study model geometries of finitely generated groups. If a finitely generated group does not contain a non-trivial finite rank free abelian commensurated subgroup, we show any model geometry is dominated by either a symmetric space of non-compact type, an infinite locally finite vertex-transitive graph, or a product of such spaces. We also prove that a finitely generated group possesses a model geometry not dominated by a locally finite graph if and only if it contains either a commensurated finite rank free abelian subgroup, or a uniformly commensurated subgroup that is a uniform lattice in a semisimple Lie group. This characterises finitely generated groups that embed as uniform lattices in locally compact groups that are not compact-by-(totally disconnected). We show the only such groups of cohomological two are surface groups and generalised Baumslag-Solitar groups, and we obtain an analogous characterisation for groups of cohomological dimension three.
