论文标题
拓扑渐近维度
Topological asymptotic dimension
论文作者
论文摘要
我们启动对局部紧凑型组的渐近维度的研究。该概念扩展了离散组的现有不变性，并且被证明是一大类残留紧凑组有限的。在此过程中，赫希长度的概念扩展到拓扑组，而赫希和马尔切夫的经典结果则使用庞加莱引理的拓扑版本扩展。我们表明，逐一压缩组并紧凑地生成，拓扑上实际上是nilpotent组是残留的，而紧凑的nilpotent组是多环状的。我们证明，对于紧凑的，可溶解的组，渐近尺寸由赫尔希长度进行了主要，而相等性则适用于多环状基团。我们将基本木材的类别延伸到离散案例之外，并表明具有有限的hirsch长度的拓扑基本的木材群具有有限的渐近维度。我们证明，拓扑基本的有限赫斯奇长度没有非平凡的本地椭圆形封闭子组是可解决的。最后，我们表明，如果其所有离散的商都是这样，则完全断开的，局部紧凑的第二个可数[sin] group具有有限的渐近维度。
We initiate a study of asymptotic dimension for locally compact groups. This notion extends the existing invariant for discrete groups and is shown to be finite for a large class of residually compact groups. Along the way, the notion of Hirsch length is extended to topological groups and classical results of Hirsch and Malcev are extended using a topological version of the Poincaré lemma. We show that polycyclic-by-compact groups and compactly generated, topologically virtually nilpotent groups are residually compact, and that compactly generated nilpotent groups are polycyclic-by-compact. We prove that for compactly generated, solvable-by-compact groups the asymptotic dimension is majorized by the Hirsch length, and equality holds for polycyclic-by-compact groups. We extend the class of elementary amenable groups beyond the discrete case and show that topologically elementary amenable groups with finite Hirsch length have finite asymptotic dimension. We prove that a topologically elementary amenable group of finite Hirsch length with no nontrivial locally elliptic normal closed subgroup is solvable-by-compact. Finally, we show that a totally disconnected, locally compact, second countable [SIN]-group has finite asymptotic dimension, if all of its discrete quotients are so.
